Gamma-radiation as a Signature of Ultra Peripheral Ion Collisions at LHC energies

0.35 MB·English

by  Yu. V. Kharlov

Nuclear γ-radiation as a Signature of Ultra
Peripheral Ion Collisions at LHC energies
Yu.V.Kharlov∗, V.L.Korotkikh†
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2 Abstract

n
a We study the peripheral ion collisions at LHC energies in which a nucleus
J
is excited to the discrete state and then emits γ-rays. Large nuclear Lorenz

3
factorallowstoobservethehighenergyphotonsuptoafewtenGeVandinthe
1
region of angles of a few hundred micro-radians around the beam direction.

1 These photons can be used for tagging the events with particle production
v
in the central rapidity region in the ultra-peripheral collisions. For that it
7
2 is necessary to have an electromagnetic detector in front of the zero degree
0 calorimeter in the LHC experiments.
1
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0 Introduction
/
h
t
-
There are several reviews devoted to the coherent γγ and γg interactions
l
c
u in the very peripheral collisions at relativistic ion colludes ([1],[2],[3]). The
n
: advantage of relativistic heavy ion colliders is that the effective photon lu-
v
i minosity for two-photon physics is of orders of magnitude higher than the
X
r one at available the e+e− machines. There are many suggestions to use the
a
electromagnetic interactions of nuclei to study production of meson reso-
nances, Higgs boson, Radion scalar or exotic mesons. These interactions
allow also to study fermion, vector meson or boson pair production, as well
as to investigate a few new physic regions (see list in [3]). The γg inter-
actions will open a new page of nuclear physics such as a study of nuclear
gluon distribution. It is also important for a knowledge of the details of
medium effects in nuclear matter at the formation of quark-gluon plasma
[4]. These effects may be studied by photo-production of heavy quarks in
virtual photon-gluon interactions ([5],[6],[4]).
For these investigations it is necessary to select the processes with large
impact parameters b of colliding nuclei, b > (R + R ), to exclude back-
1 2
ground from strong interactions. Note, that some processes, like γγ-fusion
∗Institute for High Energy Physics, 142281 Protvino, Russia
†Scobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia
1
to Higgs boson or Radion scalar, are free from any problems caused by
strong interactions of the initial state [7]. Therefore we need an efficient
trigger to distinguish γγ and γg interactions from others. G.Baur et al. [8]
suggested to measure the intact nuclei after the interaction. Evidently this
is impossible in the LHC experiments since the nuclei fly into the beam
pipe.
It is interesting to consider a γ rays emitted by the relativistic nuclei at
LHC energies. Such kind of process was used for the possible explanation
of the high energy (E ≥ 1012 eV) cosmic photon spectrum [9].
γ
It was suggested to measure a nuclear γ radiation after the excitation of
discrete nuclear level in our work [10]. These secondary photons have the
energy of a few GeV and the narrow angular distribution near the beam
direction due to a large Lorentz boost. The angular width is enough to
register them in the electromagnetic zero-degree detectors of the future
LHC experiments CMS or ALICE. A nucleus saves its Z and A in this
process. So we have a clear electromagnetic interaction of nuclei at any
impact parameter. The nuclear γ radiation may be used as “event-by-
event” criteria for such kind of collisions.
We have considered [10] only the process A+A → A∗+A+e+e−,A∗ →
A+γ′, where a nucleus is excited by electron (positron) e±+A → e±′+A∗.
Now we calculate the production process of some system X in γγ fusion
f
with simultaneous excitation of discrete nuclear level.
In this work we consider the processes
16O +16 O →16 O +16 O∗(2+,6.92 MeV) + X ,16O∗ →16 O + γ,
f
208Pb +208 Pb →208 Pb +208 Pb∗(3−,2.62 MeV) + X ,208Pb∗ →208 Pb + γ,
f
where the 16O and 208Pb were taken since they are the lightest and heaviest
ions in the ion list of the LHC program. The trigger requirements will
include a signal in the central rapidity region of particles from X decay, a
f
signal of photons in the electromagnetic detector in front of the zero degree
calorimeter and a veto signal of neutrons in ZDC. We suggest to use the
veto signal of neutrons in order to avoid the processes with the nuclear
decay into nucleon fragments.
The formalism of the considered process is presented in the section 1.
The nuclear form factors are calculated in the section 2. The angular and
energy distributions of secondary photons are in the section 3. The cross
sections of η (2.979 GeV) production are presented in the part 6 with and
c
without nuclear excitation. The section 6 is our conclusion.
2
1 Formulae of nuclear excitation cross-section and
photon luminosity in peripheral interactions
Let us consider the peripheral ion collision
A + A → A∗(λP,E ) + A + X , (1)
1 2 1 0 2 f
where X is the produced system in γ∗γ∗ fusion and A∗ is an excited
f 1
nucleus in a discrete nuclear level with spin-parity λP and energy E (see
0
Fig.1). Here the nucleus A and A have equal mass A and charge Z, the
1 2
only nucleus A is excited. We suppose that the particles of X decay
1 f
can be registered in the central rapidity region. The nuclear γ ′ radiation
(A∗ → A + γ ′) will be measured in the forward detectors such as ZDC.
1 1
Figure 1: Diagram of the process A +A → A∗(λP,E )+A +X , A∗ → A +γ.
1 2 1 0 2 f 1 1
We use the quantum mechanical plane wave formalism ([11],[3]) and
the derivation of the equivalent photon approximation. It allows us to
introduce the elastic and inelastic nuclear form factors for the process (1).
We take the formulae (19) and (21) in [3] :
dw dw
1 2
dσ = n (w )n (w )dσ (w ,w ), (2)
A1A2→A∗1A2Xf w w 1 1 2 2 γγ→Xf 1 2
Z 1 Z 2
α 1 w2m2
n (w ) = d2q dν 2 i i W + q2 W , (3)
i i π2 i⊥ i(q2)2  P2 i,1 i⊥ i,2
Z Z i i
 
where W and W are the Lorentz scalar functions. All kinematic vari-
i,1 i,2
ables are the same as in [3].
For “elastic” photon process A A → A A X we have
1 2 1 2 f
W = 0, W (ν,q2) = Z2F2(−q2)δ(ν + q2/2m) (4)
1 2 el
3
Z2α q2
n(w) = d2q ⊥ F2(−q2), (5)
π2 ⊥(q2)2 el
Z
where F (q) is the nuclear form-factor with F (0) = 1.
el el
For the excitation of nucleus to a discrete state with a spin λ and an
energy E (“inelastic” photon process A A → A∗(λP,E )A X )
0 1 2 1 0 2 f
W (ν,q2) = Wˆ (q2)δ(ν − E ),
1,2 1,2 0
w2 wE E2
−q2 = + 2 0 + 0 + q2 = q2(w) + q2,
γ2 γ γ2 ⊥ L ⊥
(6)
Wˆ = 2π[|Te|2 + |Tm|2],
1
q4 E2 − q2
Wˆ = 2π 2|Mc|2 − 0 (|Te|2 + |Tm|2) .
2 (E2 − q2)2  q2 
0
 
See notations again in [3].
We neglect the transverseelectric Te and transverse magneticTm matrix
elements comparing with the Coulomb one Mc ≡ M for 0+ → λP nuclear
λ
transitions. Then for the inelastic photon process with a nuclear discrete
state excitation we get
4α q2
n(λ)(w) = d2q ⊥ |M (q)|2, (7)
1 π ⊥(E2 − q2)2 λ
Z 0
where M (q) is the inelastic nuclear form-factor.
λ
The equivalent photon number (7) can be represented as the function
of q for inelastic photon emission:
⊥
dN(λ) 4α q2
1 (w ,q ) = ⊥ |M (−q2)|2 =
dq2 1 ⊥ π (E2 − q2)2 λ
⊥ 0
4α q 2
= ⊥ M (−q2)eiϕ⊥ , (8)
λ
π (cid:12)(E2 − q2) (cid:12)
(cid:12) 0 (cid:12)
(cid:12) (cid:12)
where q⊥eiϕ⊥ = q~⊥ (see [12]). (cid:12)(cid:12) (cid:12)(cid:12)
Let us do the inverse transformation to the impact parameter b presen-
tation
1
f(~b) = d2q e−i~q⊥~bf(~q ). (9)
⊥ ⊥
2π
Z
For the function under the module in equation (8) we get
1 q
f(~b) = d2q ⊥ M (−q2)eiϕ⊥ · e−i~q⊥~b =
2π ⊥(E2 − q2) λ
Z 0
q2
= i dq ⊥ M (−q2) · J (q b) =
⊥(E2 − q2) λ 1 ⊥
Z 0
i u2 x2 + u2
= du M − J (u). (10)
b u2 + (E2 + q2) b2 λ b2  1
Z 0 L
 
4
If we take M instead of the inelastic M as
el λ
Z2
|M (−q2)|2 = F2(−q2) (11)
el el
4π
we get a well-known formula for elastic photon process (see (4) in [12])
where F (0) = 1:
el
Z2α 1 u2 2
N(el)(w,b) = du J (u)F [−(x2 + u2)/b2] , (12)
2 π2 b2(cid:12) x2 + u2 1 el (cid:12)
(cid:12)Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Here x = q b = wb/γ an(cid:12) d u = q b. For a point charge, F (cid:12)(q) ≡ 1, we
L A ⊥ el
readily obtain
Z2α 1
N(el)(w,b) = x2 K2(x), (13)
2 π2 b2 1
in agreement with [3] at very large γ .
A
We write the form factors of elastic and inelastic nuclear process in the
same forms:
1
F2(q) = F2(q) (14)
λ 4πe2Z2 λ
1 2
F2(q) = 4π sin(qr)ρ (r)rdr → 1, (15)
0 (cid:12) q 0 (cid:12)
(cid:12) Z (cid:12)q→0
(cid:12) (cid:12)
(cid:12) (cid:12) 2
F2(q) = (cid:12)(2λ + 1) 4π j (qr)ρ ((cid:12)r,Z)r2dr → (16)
λ λ λ
(cid:12) (cid:12)
(cid:12) Z (cid:12)q→0
(cid:12) (cid:12)
(4π)2B(cid:12) (Eλ) (cid:12)
(cid:12) (cid:12)
→ q2λ, (17)
e2Z2[(2λ+ 1)!!]2
where ρ (r,Z) is an nuclear transition density and B(E λ) is the reduced
λ 0
transition probability .
Then for the matrix elements M we get in the limit q → 0
λ
Z2 Z2
|M (−q2)|2 = F2(q) → (18)
el 4π el (cid:12) 4π
(cid:12)
(cid:12)q→0
  (cid:12)
(cid:12)
Z2 Z2 (cid:12) (4π)2B(E λ)
|M (−q2)|2 = F2(q) → 0 q2λ. (19)
λ 4π λ (cid:12) 4π e2Z2[(2λ+ 1)!!]2
(cid:12)
(cid:12)q→0
  (cid:12)  
The effective photon number(cid:12)for inelastic process with nuclear transition
(cid:12)
0 → λ will be
Z2α 1 ∞ u2 2
N(λ)(w,b) = du J (u)F [−(x2 + u2)/b2] , (20)
1 π2 b2(cid:12) x2 + u2 1 λ in (cid:12)
(cid:12)Z0 in (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
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as the generalization of (12). Here x2 = (E2 + w2 + 2wE0 + E02) b2.
in 0 γ2 γ γ2
We take the inelastic form-factor from inelastic electron scattering off
nuclei. A good parameterization of inelastic form-factor is
−q2g2
F2(q) = 4πβ2j2(qR)e (21)
λ λ λ
in the Helm’s model [13]. The squared transition radius is equal to R2 =
λ
R2 + (2λ + 3)g2, where g is a size of a nuclear diffusion side.
The reduced transition probability in this case is equal to
β2
B(E λ) = λZ2e2R2λ. (22)
0
4π
So, the formulae for the process (1) are
dw dw
1 2 (λ)
dσ = n (w )n (w )dσ (w ,w ); (23)
A1A2→A∗1A2Xf w w 1 1 2 2 γγ→Xf 1 2
Z 1 Z 2
Z2α q2
n(λ)(w ) = d2q ⊥ |F (−q2 )|2; (24)
1 1 π2 ⊥(E2 − q2 )2 λ in
Z 0 in
w2 wE E2
−q2 = + 2 0 + 0 + q2; (25)
in γ2 γ γ2 ⊥
A A A
Z2α q2
n (w ) = d2q ⊥F2(−q2); (26)
2 2 π2 ⊥q4 el el
Z el
w 2
−q2 = + q2. (27)
el γA! ⊥
The value q2 is close to q2 at a large γ factor at LHC energies.
in el A
The effective two photon luminosity can be expressed as
∞ ∞ 2π
L(ω ,ω ) = 2π b db b db dφN(λ)(ω ,b )N(el)(ω ,b )Θ(B2), (28)
1 2 1 1 2 2 1 1 1 2 2 2
Z Z Z
R1 R2 0
where R and R are the nuclear radii, Θ(B2) is the step function and
1 2
B2 = b2 + b2 − 2b b cosφ − (R + R )2 [3]. Then the final cross-section is
1 2 1 2 1 2
dω dω
1 2
σ = L(ω ,ω ) σ (w ,w ) (29)
A1A2→A∗1A2Xf ω ω 1 2 γγ→Xf 1 2
Z 1 Z 2
2 Nuclear levels and form-factors
The elastic form factor of a light nucleus is
hr2i
F (q2) = exp − q2 (30)
el
 6 
 
6
with hr2i = 2.73 fm for the nucleus 16O. For a heavy nucleus we take a
modifiqed Fermi nuclear density [14]
1 1
ρ(r) = ρ + − 1 (31)
0 1 + exp−r−R 1 + expr−R 
 g g 
sh(R/g)
 
= ρ , (32)
0
ch(R/g) + ch(r/g)
3 πg 2 −1
ρ = 1 + (33)
0
4πR3  R ! 
 
with the parameters for 208Pbare equal toR = 6.69 fm,g = 0.545 fm. Such
form of density is close to the usual Fermi density at g ≪ R
1
ρ (r) = ρ (34)
F 0 1 + exp r−R
g
and allows us to calculate analytically the elastic form factor
4π2Rgρ πg
0
F (q) = sin(qR) cth(πgq) − cos(qR) . (35)
el
q sh(πgq) ( R )
There are a few discrete levels of 16O below α, p and n thresholds
E (α) = 7.16 MeV, E (p) = 7.16 MeV, E (n) = 7.16 MeV [15]. The
th th th
level 2+ at E = 6.92 MeV is the strongest excited one in the electron
0
scattering.
The parameters from the inelastic electron scattering fit on 16O with
excitation of 2+ level (E = 6.92 MeV) of 16O are [16]:
0
β = 0.30, R = 2.98 fm, g = 0.93 fm.
2
They correspond to
B(E 2) = (36.1± 3.4)e2 fm4. (36)
0
There are more than 70 discrete levels of 208Pb [17] below the neutron
thresholdE (n) = 7.367MeV. About 30% of the levelsdecay to the first 3−
th
level of 208Pb at E = 2.615 MeV. This level is well studied experimentally
0
[18] and has a large excited cross-section.
The reduced transition probability from the fit of inelastic electron scat-
tering on 208Pb with excitation of the 3− level is [18]:
B(E 3) = (6.12 105 ± 2.2%)e2 fm6. (37)
0
7
We calculate the parameter β , using this B(E 3), and take R and g
3 0
from the density of the 208Pb ground state:
β = 0.113, R = 6.69 fm, g = 0.545 fm.
3
Note that there are many levels higher than E = 2.615 MeV which
0
decay to the first level of 208Pb. This fact increases the event rate of the
process (1), but we don’t know cross-section excitation of these levels.
The elastic form factor (30) of 16O and inelasticform-factor 16O (2+,6.92
MeV) (21), corresponding to the electron scattering data, are shown in
Fig.2. Thesame fora nucleus 208Pb and theexitedstate208Pb(3−,2.64MeV)
are shown in Fig.3.
1
1
10-1
10-2
)
q 2
2( F10-3
10-4
10-5
10-6
0 0.5 1 1.5 2 2.5
-1
q, fm
Figure 2: The elastic form-factor of 16O (1) and the inelastic form-factor of 16O (2+,6.92
MeV) (2) from the electron scattering.
The squared inelastic form-factor is less than the elastic form-factor by
more then two orders at small q < q (q = 1 fm−1 for 16O and q = 0.6 fm−1
0 0 0
for 208Pb). In the region of q ≃ q they are comparable. The region of
0
large q > q will give contribution for the small impact parameter b. We
0
are able to calculate the photon luminosity (28) for all regions of b to get
the maximum electromagnetic cross-section of process we are interested
in. Then it should be possible to compare with experimental data in the
condition of clear selection of such process by the photon signal and the
veto neutron or proton signal in ZDC.
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1
1
10-1
10-2
2
)q 10-3
2( F
10-4
10-5
10-6
0 0.2 0.4 0.6 0.8 1 1.2 1.4
-1
q, fm
Figure 3: The elastic form-factor of 208Pb (1) and the inelastic form-factor of
208Pb (3−,2.615 MeV) (2).
3 Angular and energy distributions of secondary nu-
clear photons
We suppose that the nucleus A∗ in the process (1) is unpolarized. Just
1
now we don’t know the relative excitation probability of |λµ > state of
A∗, where µ is a projection of spin λ. This assumption needs the study in
1
future. So we use a formula (27) in our work [10] for the angular distribu-
tion of secondary photons, which is valid for isotropic photon distribution
in the rest system of A∗.
1
If we know the cross-section of reaction (1) calculated by the equation
(29) then the angular and energy distribution of photons are equal to:
dσ 2γ2 sinθ
A∗ A γ
= σ · . (38)
dθ A1A2→A∗1A2X (1 + γ2 tan2θ )2 · cos3θ
γ A∗ γ γ
1
The photon energy E and polar angle θ in laboratory system are
γ γ
defined as:
E = γ E (1 + cosθ′ ) = 2γ E /(1 + γ2 tan2θ ), (39)
γ A∗1 0 γ A∗1 0 A∗1 γ
1 sinθ′
γ
tanθ = , (40)
γ
γ 1 + cosθ′
A∗1 γ
where θ′ and θ are polar angles of nuclear photon in the rest nuclear
γ γ
system and in the laboratory system with an axis ~z||p~ . Photon energy
A∗
E dependence on θ are shown in Fig.4.
γ γ
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Our calculations with the help of TPHIC event generator [19] show that
a deflection of the direction p~ from ~p at LHC energies in the reaction
A∗ beam
(1) is very small at large γ , h∆θi ≃ 0.5 µrad.
A
60
50
40
30
20
10
9
8
7
6
0 100 200 300 400 500
Figure 4: Nuclear photon energy as function of its polar angle in the laboratory system
at LHC energies for two nuclei: 160 (2+ → 0+,6.92 MeV) (1) and 208Pb (3− → 0+,2.615
MeV) (2). ZDC marks a region of Zero Degree Calorimeter in CMS.
In theexperimentsCMS and ALICE, whichareplannedat LHC (CERN),
the Zero Degree Calorimeter ([20], [21]) were suggested for the registra-
tion of nuclear neurons after interaction of two ions. We demonstrate a
schematic figure of ZDC (CMS) at a distance L = 140 m in the plane trans-
verse to the beam direction in Fig.5. The CMS group plans to include also
the electromagnetic calorimeter in front of ZDC.
As an example we demonstrate the angular distributions (38) in arbi-
trary units and energy dependence (39) on the (x,y) coordinates of ZDC
(CMS) for two nuclei 16O and 208Pb in Fig.6. The direction of the nucleus
A∗ coincides here the beam direction. A point (x,y) = (0,0) is a center of
1
the ZDC plane.
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