PHYSICAL REVIEW D 103, 033007 (2021) - Exploring the double-slit interference with linearly polarized photons
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PHYSICAL REVIEW D 103, 033007 (2021)
Exploring the double-slit interference with linearly polarized photons
Wangmei Zha ,1 James Daniel Brandenburg ,2,3,* Lijuan Ruan,3 and Zebo Tang 1,†
1
University of Science and Technology of China, 230026 Hefei, China
2
Shandong University, 250100 Jinan, China
3
Brookhaven National Laboratory, New York 11973, USA
(Received 2 July 2020; revised 5 November 2020; accepted 19 January 2021; published 26 February 2021)
The linearly polarized quasireal photons from the highly Lorentz-contracted Coulomb fields of
relativistic heavy ions can fluctuate to quark-antiquark pairs, scatter off a target nucleus, and emerge
as vector mesons. In the process, the two colliding nuclei can serve as slits, forming a Young’s double-slit
experiment. In addition to the well-known double-slit interference pattern in position space, a similar
interference pattern may be expected in polarization space due to the linear polarization of the colliding
photons. In this paper, we investigate the interference effect in polarization space as revealed by the
asymmetries of the decay angular distribution for vector meson photoprodution in heavy-ion collisions.
We find a periodic oscillation pattern with transverse momentum, which can reasonably explain the
second-order modulation in azimuth for the ρ0 decay observed by the STAR Collaboration.
DOI: 10.1103/PhysRevD.103.033007
The double-slit interference experiment plays an essen- decay products in an entangled state, which is an illus-
tial role in expressing the central puzzle of quantum tration of the Einstein-Podolsky-Rosen paradox. The dou-
mechanics—wave-particle complementarity. Such wave- ble-slit interference effect of photonuclear vector meson
particle duality continues to be challenged [1–5] and production in momentum space has been proposed by
explored in a broad range of entities [6–11]. The dou- Klein and Nystrand [13] and verified by the STAR
ble-slit experiment could also be performed in relativistic Collaboration through measurements of the ρ0 in ultra-
heavy-ion collisions utilizing the vector meson from peripheral collisions (UPCs) [14]. In Refs. [15,16], we
coherent photoproduction. The highly Lorentz-contracted extended the scenario to hadronic heavy-ion collisions and
electromagnetic field accompanied by the heavy nuclei can proposed to exploit the strong interactions in the overlap
be viewed as a spectrum of quasireal photons [12]. The region to test the observation effect on the interference.
photon emitted from one nucleus can fluctuate to a quark- The photons generated from the highly Lorentz-
antiquark pair which then scatters off the other nucleus, contracted electromagnetic field are expected to be fully
emerging as a real vector meson. In the production process, linearly polarized. It has been suggested by Li et al. [17]
the elastic scattering occurs via Pomeron exchange, which that the collisions of the linearly polarized photons
imposes a restriction on the production site within one of (γ þ γ → lþ þ l− ) result in the second-order and fourth-
the two colliding nuclei. This indicates that the production order modulations in azimuth (in the plane perpendicular to
source consists of two well-separated nuclei (two slits). the beam direction) between the pair momentum and the
There are two possibilities: Either nucleus 1 emits a photon lepton momentum for dilepton production. The experimen-
and nucleus 2 acts as a target, or vice versa. The two tal signature was confirmed by the STAR Collaboration
possibilities are indistinguishable, which present the pro- for the dielectron measurements [18]. The vector meson
duction process as a perfectly double-slit experiment. production from linearly polarized photons also possesses
Furthermore, for the short-lived vector meson (e.g., ρ0 ), a distinctive signature in the asymmetries of the decay
it decays before the wave functions from the two slits can angular distributions. Continuous efforts [19–26] has been
overlap, and the interference entity is the complete set of made for more than 50 years to utilize the linearly polarized
photons in vector meson photoproduction as a parity filter
*
dbrandenburg.ufl@gmail.com for the exchange of particles in the t channel, which is an
†
zbtang@ustc.edu.cn effective tool to separate the natural from unnatural parity
exchange in the t channel. Since the photons are in a linear
Published by the American Physical Society under the terms of polarization state, the interference can reveal itself in
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to polarization space, inducing an asymmetry in the decay
the author(s) and the published article’s title, journal citation, angular distributions. In this paper, we exploit the linearly
and DOI. Funded by SCOAP3. polarized photons to investigate the interference pattern in
2470-0010=2021=103(3)=033007(6) 033007-1 Published by the American Physical SocietyZHA, BRANDENBURG, RUAN, and TANG PHYS. REV. D 103, 033007 (2021)
z
z'
V rest frame
π
θ
ϕ
Φ
x y
x'
FIG. 1. The coordinate system for the measurement of a two- FIG. 2. Schematic diagram for the direction of electric vector of
body decay angular distribution in the vector meson rest frame. the photons, which hit on the target nuclei, in ultraperipheral
The definition of the x, y, and z axes is described in the text. The heavy-ion collisions.
z0 axis denotes the beam direction, and the x0 axis represents the
linear polarization vector. and to spin 1=2 products (e.g., ρ0 → eþ þ e− ) gives
d2 N 3 2 sin2 θ
polarization space for the vector meson photoproduction in ¼ ð1 þ cos θÞ 1 − cos 2ðϕ − ΦÞ :
heavy-ion collisions and demonstrate that the signature is a d cos θdϕ 16π 1 þ cos2 θ
periodic oscillation in asymmetries of the decay angular ð2Þ
distributions with transverse momentum.
Hereinafter, we take the process of γ þ A → V þ A → As revealed in Eqs. (1) and (2), the linearly polarized states
dþ þ d− þ A in heavy-ion collisions to illustrate the result in the second-order modulations in azimuth, and the
interference effect on the asymmetries of the decay. The strength of the second-order modulation is given by
decay distribution depends on the choice of the coordinate
system, with respect to which the momentum of one of the 2hcosð2ϕÞi ¼ cosð2ΦÞ ð3Þ
two decay products is expressed in spherical coordinates.
Herein, the right-handed coordinate system for vector and
meson decay is built up as follows: The z axis is chosen
to the direction of flight of the vector meson in the photon- sin2 θ
2hcosð2ϕÞi ¼ − cosð2ΦÞ; ð4Þ
nucleon center of mass frame; the y axis is normal to the 1 þ cos2 θ
photoproduction plane; and the x axis is given by y × z. As
illustrated in Fig. 1, the decay angles θ and ϕ are the polar for the ρ0 → π þ þ π − and ρ0 → eþ þ e− cases, respec-
and azimuthal angles, respectively, of the unit vector π̃, tively. The modulation strength is determined by the
which denotes the direction of flight of one of the decay direction of the linear polarization, and the orientation of
particles in the vector meson rest frame. In experiment, the the photon polarization is determined by the direction of the
direction of the z axis is approximated by that of incoming electric field vector. As demonstrated in Fig. 2, the electric
beams (shown as the z0 axis in Fig. 1). It has been verified field vector is parallel to the impact parameter at leading
by Monte Carlo calculations that this is a good approxi- order. That is to say that the modulations of the decay
mation. Here, we adopt the approximation in our calcu- distribution for vector meson are determined by the
lation for direct comparisons with experimental results. anisotropy in its two-dimensional transverse momentum
The angle Φ shown in Fig. 1 denotes the angle between distribution with respect to the impact parameter at leading
the photon polarization plane and vector meson produc- order. The variation of polarization direction due to the
tion plane. finite size of nuclei should be small in ultraperipheral
Under the helicity no-flip assumption, the vector meson collisions, which could be investigated in future work.
inherits the photon polarization state, which is fully linearly The two-dimensional transverse momentum distribution
polarized. The helicity conservation assumption has of the vector meson from coherent photoproduction can be
been investigated by various experimental measurements obtained by performing a Fourier transformation of the
[20–25]. Following Ref. [27] and the derivation in the coordinate space amplitude:
Appendix, the decay angular distribution of vector meson Z 2
d2 P 1
to two spinless products (e.g., ρ0 → π þ þ π − ) is ¼ 2
d x⊥ ðA1 ðx⊥ Þ þ A2 ðx⊥ ÞÞeip⊥ ·x⊥
ð5Þ
dpx dpy 2π ;
d2 N 3 where A1 ðx⊥ Þ and A2 ðx⊥ Þ are the amplitude distributions in
¼ sin2 θ½1 þ cos 2ðϕ − ΦÞ; ð1Þ
d cos θdϕ 8π the transverse plane for the two colliding nuclei. Consider,
033007-2EXPLORING THE DOUBLE-SLIT INTERFERENCE WITH … PHYS. REV. D 103, 033007 (2021)
(a) (b) (c)
A 1 = A2 A1 = 0.5A2
0.4 0.4
0.2 0.2
2⟨cos2φ⟩
2⟨cos2φ⟩
0 0
−0.2 −0.2
−0.4 −0.4
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
p GeV/c p GeV/c
T T
FIG. 3. (a) Two-dimensional transverse momentum distribution pattern with pointlike assumption at midrapidity. (b) The modulation
strength 2hcos ϕi as a function of transverse momentum at midrapidity. (c) 2hcos ϕi as a function of transverse momentum at forward
rapidity (A1 ≠ A2 ).
to leading order, we can approximate the production photon approximation, the spatial photon flux generated by
over the two nuclei as two point sources: A1;2 ðx⊥ Þ ¼ the heavy nuclei can be written as
A1;2 δðx⊥ 12 bÞ, where A1 and A2 are the relative ampli- Z
tudes for the two colliding nuclei, δ is the Dirac δ function, 4Z2 α d2 k⃗ γ⊥ ⃗ Fγ ðk⃗ γ Þ i⃗x ·k⃗ 2
nðω; x⃗ ⊥ Þ ¼ k e ⊥ γ⊥
;
b is the impact parameter, and the minus sign in A2 ðx⊥ Þ ωγ ð2πÞ2 γ⊥ jk⃗ γ j2
results from the negative parity of the vector meson. With
the interference effect, as demonstrated in Fig. 3(a) for the ⃗kγ ¼ k⃗ γ⊥ ; ωγ ; 1
ωγ ¼ MV ey ; ð6Þ
case with b ¼ 20 fm at midrapidity (A1 ¼ A2 ), the trans- γc 2
verse momentum reveals a typical Young’s double-slit
interference pattern with a series of alternating light and where x⃗ ⊥ and k⃗ γ⊥ are two-dimensional photon position and
dark fringes, which corresponds to significant asymmetries momentum vectors, respectively, perpendicular to the beam
in azimuth. According to Eq. (3), the modulation strength direction, ωγ is the energy of the emitted photon, Z is the
factor of vector meson decay to two spinless products, electric charge of the nucleus, α is the electromagnetic
2hcosð2ϕÞi, versus transverse momentum (pT ) can be coupling constant, γ c is the Lorentz factor of the beam, MV
extracted from the two-dimensional distribution. The result and y are the mass and rapidity, respectively, of the vector
is shown in Fig. 3(b). The modulation strength shows a meson, and Fγ ðk⃗ γ Þ is the nuclear electromagnetic form
periodic oscillation like the hyperbolic Bessel functions factor. The form factor can be obtained via the Fourier
with the transverse momentum. The oscillation cycle is transformation of the charge density in the nucleus. The
determined by the distance between the two source (impact charge density profile of the nucleus can be parametrized
parameter). For production at forward rapidity in heavy-ion by the Woods-Saxon distribution:
collisions, the relative amplitudes from the two nuclei are
not equal. Figure 3(c) shows the modulation strength as a a0
function of transverse momentum with unequal amplitudes ρA ðrÞ ¼ ; ð7Þ
1 þ exp½ðr − RWS Þ=d
from the two sources. Since the interference is not complete
in this case, the oscillation level is weaker than that at where the radius RWS and skin depth d are based on fits
midrapidity. Furthermore, the modulation strength is 0 at to electron-scattering data [28] and a0 is the normali-
pT ¼ 0, while it is maximum for the case at midrapidity. zation factor. As demonstrated in Eq. (6), the polarization
This results since the interference contribution is close to 0 direction for the quasireal photons follows the position
for pT ¼ 0. vector x⃗ ⊥.
As mentioned herein before, we have qualitatively The scattering amplitude ΓγA→VA with the shadowing
demonstrated the interference effect on the asymmetries effect can be estimated via the Glauber [29] þ vector meson
of the decay angular distributions for vector meson photo- dominance (VMD) approach [30]:
production. The quantitative calculations are described
hereinafter. Following Ref. [15], the production amplitude
f γN→VN ð0Þ σ VN 0
distribution for vector meson photoproduction is deter- ΓγA→VA ð⃗x⊥ Þ ¼ 2 1 − exp − T ð⃗x⊥ Þ ;
σ VN 2
mined by the spatial photon flux and the corresponding
γA scattering amplitude ΓγA→VA . According to equivalent ð8Þ
033007-3ZHA, BRANDENBURG, RUAN, and TANG PHYS. REV. D 103, 033007 (2021)
where f γN→VN ð0Þ is the forward-scattering amplitude for where σ NN is the total nucleon-nucleon cross section. Then,
γ þ N → V þ N and σ VN is the total VN cross section. To the probability of having no hadronic interaction is
account for the coherent length effect, the modified thick- exp½−PH ðbÞ. In experiment, the UPC events are usually
ness function T 0 ð⃗x⊥ Þ is defined as triggered or selected by the mutual Coulomb dissociation
of the nuclei, in which the number of emitted forward
Z þ∞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M V ey neutrons can be detected by zero degree calorimeters.
T 0 ð⃗x⊥ Þ ¼ ρ x⃗ 2⊥ þ z2 eiqL z dz; qL ¼ ;
−∞ 2γ c Following the EPA approach, the Coulomb excitation
probability of an ultrarelativistic nucleus can be determined
ð9Þ
by the photon flux accompanied with the nucleus and the
appropriate photon-absorption cross section of nuclei. The
where qL is the longitudinal momentum transfer required lowest-order probability for an excitation of nucleus which
to produce a real vector meson. The f γN→VN ð0Þ can be emits at least one neutron (Xn ) is
determined from the measurements of forward-scattering
dσ Z
cross section γN→VN jt¼0 , which is well parametrized in
dt
mXn ðbÞ ¼ dωnðω; bÞσ γA→A ðωÞ; ð14Þ
Ref. [31]. Using the optical theorem and VMD relation, the
total cross for VN scattering can be given by
where nðω; bÞ is photon flux described by Eq. (6) and
σ VN
f
¼ pVffiffiffi f γV→VN ; ð10Þ σ γA→A ðωÞ is the photoexcitation cross section with incident
4 αC energy ω. The photoexcitation cross section σ γA→A ðωÞ can
be extracted from the experimental measurements [33–39].
where f V is the V-photon coupling and C is a correction The probability of mutual dissociation for the two nuclei
factor for the nondiagonal coupling through higher mass with at least one neutron emission for each beam (XnXn) is
vector mesons. Finally, the production amplitude can be then given by
given by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PXnXn ðbÞ ¼ ð1 − exp½−mXn ðbÞÞ2 : ð15Þ
Aðx⃗ ⊥ Þ ¼ ΓγA→VA nðω; x⃗ ⊥ Þ: ð11Þ
As described in our recent work [40], the mutual dissoci-
In addition to the coherent production with interference, ation probability with any number of neutron emission can
there are also an incoherent contribution in heavy-ion be estimated in a similar way.
collisions, which is significant at relative high transverse With the inputs described before, the cross section and
momentum. The incoherent production should have no differential distributions for vector meson photoproduction
contribution to the asymmetries of decay distribution, can be calculated and can reasonably describe the mea-
which would weaken the overall modulation strength. surements from STAR [24,41] and ALICE [42,43]. The
Similarly to the case of coherent vector meson photo- estimated modulation strength 2hcosð2ϕÞi of ρ0 → π þ þ
production on nuclei, the incoherent cross section σ γA→VA0 π − and ρ0 → eþ þ e− as a function of pT in ultraperipheral
could be related to the exclusive γp cross section σ γp→Vp pffiffiffiffiffiffiffiffi
Au þ Au collisions at sNN ¼ 200 GeV for mutual dis-
via the Glauber þ VMD approach. Here, A0 is the final sociation mode XnXn are shown in Fig. 4. The modulation
nuclear state containing products of the nuclear disintegra- strength shows a periodic oscillation with the transverse
tion. The approach gives momentum. The amplitude of the second oscillation period
Z is weaker than that of the first oscillation period, which is
σ γA→VA0 ¼ σ γp→Vp d2 x⃗ ⊥ Tð⃗x⊥ Þe−ð1=2Þσ VN Tð⃗x⊥ Þ ;
in
ð12Þ due to a larger fraction of incoherent production at higher
pT . The oscillation feature completely disappears for
pT > 0.16 GeV=c, where the incoherent production plays
where σ in 2
VN ¼ σ VN − σ VN =ð16πBV Þ is the inelastic vector the dominate role in the photoproduction. As expected in
meson-nucleon cross section and BV is the slope of the t Eqs. (3) and (4), the modulation strength for ρ0 → eþ þ e−
dependence of the γp → Vp scattering [31]. is weaker and possesses opposite sign in comparison with
Here, we focus on the ultraperipheral collisions, where that for ρ0 → π þ þ π −. Furthermore, it depends on the
there is no nuclear overlap to reject hadronic background. acceptance, and the calculations for ρ0 → eþ þ e− are
According to the optical Glauber model [32], the mean performed with STAR acceptance as listed in the figure.
number of projectile nucleons that interact at least once in The results for ρ0 → π þ þ π − can reasonably describe
A þ A collisions with impact parameter b is the transverse momentum dependence of the second-
Z order modulation in azimuth observed by the STAR
PH ðbÞ ¼ ⃗
d2 x⃗ ⊥ Tð⃗x⊥ − bÞf1 − exp½−σ NN Tð⃗x⊥ Þg; ð13Þ Collaboration [44]. The predictions for ρ0 → eþ þ e− call
for further experimental testing.
033007-4EXPLORING THE DOUBLE-SLIT INTERFERENCE WITH … PHYS. REV. D 103, 033007 (2021)
0.5
jVi ¼ aþ1 j þ 1i þ a−1 j − 1i þ a0 j0i: ðA1Þ
0.4
photoproduction ρ0 → π+ + π-
Au + Au 200 GeV ρ0 → e+ + e-
0.3
The calculation is performed in the vector meson rest
XnXn |y|0.2 GeV/c |ηe|ZHA, BRANDENBURG, RUAN, and TANG PHYS. REV. D 103, 033007 (2021)
of a vector meson to two spinless products (e.g., Wðcos θ; ϕÞ
ρ0 → π þ þ π − ), with d100 ¼ cos θ and d110 ¼∓ p1ffiffi2 sin θ, X
∝ jBl0 j2
the decay distribution can be written as l0 ¼1
3 3
Wðcos θ; ϕÞ ∝ jB0 j2 ∝ sin2 θ½1 þ cos 2ðϕ − ΦÞ: ðA7Þ ∝ ½1 þ cos2 θ − sin2 θ cos 2ðϕ − ΦÞ
8π 16π
3 sin2 θ
For the decay to a dilepton system (e.g., ρ0 → eþ þ e− ), ∝ ð1 þ cos2 θÞ 1 − cos 2ðϕ − ΦÞ : ðA8Þ
16π 1 þ cos2 θ
with d101 ¼ p1ffiffi2 sinθ, d111 ¼ 1þcosθ 1 1−cosθ
2 , and d∓11 ¼ 2 ,
the decay distribution can be written as
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