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Eur. Phys. J. A 21, 437­443 (2004) DOI 10.1140/ep ja/i2004-10008-x

THE EUROPEAN PHYSICAL JOURNAL A

Nuclear -radiation as a signature of ultra-p eripheral ion collisions at the LHC
Yu.V. Kharlov1 and V.L. Korotkikh
1 2

2,a

Institute for High Energy Physics, 142281 Protvino, Russia Scobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia Received: 10 February 2004 / Revised version: 9 March 2004 / Published online: 14 September 2004 ­ c Societ` Italiana di Fisica / Springer-Verlag 2004 a Communicated by W. Henning Abstract. We study the peripheral ion collisions at LHC energies where a nucleus is excited to a discrete state and then emits -rays. Large nuclear Lorenz factors allow the observation of high-energy photons up to a few tens GeV and in the angular region of a few hundred micro-radians from the beam direction. These photons can be used to tagg events with particle production in the central rapidity region in ultraperipheral collisions. To detect these photons it is necessary to have an electromagnetic detector in front of the zero-degree calorimeter in LHC experiments. PACS. 25.75.-q Relativistic heavy-ion collisions ­ 25.75.Dw Particle and resonance production

Intro duction
There are several reviews devoted to coherent and g interactions in very peripheral collisions at relativistic ion colliders [1­3]. The advantage of relativistic heavy-ion colliders is that the effective photon luminosity for twophoton physics is orders of magnitude higher than that available in e+ e- machines. There have been many suggestions to use the electromagnetic interactions of nuclei to study production of meson resonances, Higgs bosons, Radions or exotic mesons. These interactions also probe fermion, vector meson or boson pair production, as well as investigate some new physics regions (see list in ref. [3]). The g interactions will open a new area of nuclear physics such as the study of nuclear gluon distribution. It is also important for the knowledge of the details of medium effects in nuclear matter at the formation of the quarkgluon plasma [4]. These effects may be studied by photoproduction of heavy quarks in virtual photon-gluon interactions [4­6]. For these investigations it is necessary to select processes with large impact parameters b of the colliding nuclei, b > (R1 + R2 ), to exclude background from strong interactions. Note that some processes, like -fusion to Higgs bosons or Radions, are free from any problems caused by strong interactions of the initial state [7]. Therefore, we need an efficient trigger to distinguish and g interactions from others. G. Baur et al. [8] suggested to detect intact nuclei after the interaction. Evidently this is
a

e-mail: vlk@lav01.sinp.msu.ru

impossible in LHC experiments since nuclei fly into the beam pipe. It is interesting to consider -rays emitted by the relativistic nuclei at LHC energies. This process was used for the possible explanation of the high-energy (E 1012 eV) cosmic photon spectrum [9]. We had considered [10] the process A + A A + A + e+ e- , A A + , where a nucleus is excited by the electron (positron) e± + A e± + A , and suggested to detect a nuclear radiation after the excitation of discrete nuclear levels [10]. These secondary photons have the energy of a few GeV and a narrow angular distribution close the beam direction due to a large Lorentz boost. The angular width is large enough for them to be detected in the electromagnetic zero-degree calorimeters (ZDC) of the future LHC experiments CMS or ALICE. Now we calculate the production process of some system Xf in fusion with simultaneous excitation of the discrete nuclear level. The nucleus retains its charge Z and mass A in this process. So we have a clear electromagnetic interaction of nuclei at any impact parameter. The nuclear radiation may be used as "event-by-event" criteria in these collisions. In this work we consider the processes 16 O +16 O 16 O +16 O (2+ , 6.92 MeV) + Xf , 16 O 16 O + , 208 208 208 Pb + Pb Pb +208 Pb (3- , 2.62 MeV) + Xf , 208 Pb 208 Pb + , 16 208 where the O and Pb were taken since they are the lightest and heaviest ions in the LHC program. The trigger requirements will include a signal in the central rapidity


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The European Physical Journal A

For the "elastic" photon process A1 A2 A1 A2 Xf we have
2 W1 = 0, W2 (, q 2 ) = Z 2 Fel (-q 2 ) ( + q 2 /2m) .

(4)

So that [3] n(w) = Z 2 2 d2 q
2 q 2 F (-q 2 ) , (q 2 )2 el

(5)

Fig. 1. Diagram of the process A1 + A2 A ( , E0 ) + A2 + Xf , A A 1 + . 1
1

P

where Fel (q ) is the nuclear form-factor with Fel (0) = 1. For the excitation of the nucleus to a discrete state with a spin and an energy E0 ("inelastic" photon process A1 A2 A (P , E0 )A2 Xf ) 1 W
1,2

region of particles from Xf decay, a signal of photons in the electromagnetic detector in front of the zero-degree calorimeter and a veto signal of neutrons in the ZDC. We suggest to use the veto signal of neutrons in order to avoid the processes with nuclear decay into nucleon fragments. The formalism of the considered process is presented in sect. 1. The nuclear form factors are calculated in sect. 2. The angular and energy distributions of secondary photons are in sect. 3. The cross-sections of c (2.979 GeV) production are presented in sect. 4 with and without nuclear excitation. Section 5 is our conclusion.

^ (, q 2 ) = W1,2 (q 2 ) ( - E0 ), 2 w E0 E0 w2 2 + 2 + q , -q 2 = 2 + 2 ^ W1 = 2 [|T e |2 + |T m |2 ], ^ W2 = 2 q4 2 (E0 - q 2 )
2 2 E0 - q 2 (|T e |2 + |T m |2 ) . (6) q2

â 2|M C |2 -

1 Formulae of nuclear excitation cross-section and photon luminosity in p eripheral interactions
Let us consider the peripheral ion collision A1 + A2 A (P , E0 ) + A2 + Xf , 1 (1)

See notations again in [3]. We neglect the transverse electric T e and transverse magnetic T m matrix elements compared to the Coulomb one M C M for 0+ P nuclear transitions. Then for the "inelastic" photon process with a nuclear discrete state excitation we get n
() 1

(w) =

4

d2 q



2 q |M (-q 2 )|2 , (E - q 2 )2 2 0

(7)

where Xf is the produced system in fusion and A 1 is an excited nucleus in a discrete nuclear state with spinparity P and energy E0 (see fig. 1). Here the nuclei A1 and A2 have equal mass A and charge Z , only the nucleus A1 is excited. We suppose that the reaction product Xf decay can be detected in the central rapidity region. The nuclear radiation A A1 + will be measured in the 1 forward detectors as ZDC. We use the quantum-mechanical plane-wave formalism [3, 11] and the derivation of the equivalent photon approximation. This allows us to introduce the elastic and inelastic nuclear form factors for process (1). We take the formulae (19) and (21) in [3] : d
A1 A2 A A2 X 1
f

where M (q ) is the inelastic nuclear form factor and -q 2 = 2 2 qL (w) + q . The equivalent photon number (7) can be represented as function of q for inelastic photon emission:
2 4 dn1 q 22 2 (w1 , q ) = (E 2 - q 2 )2 |M (-q )| = dq 0 ()

4 q = M (-q 2 )e 2 (E0 - q 2 )

2 i


,

(8)

where q ei = q (see [12]). Let us do the inverse transformation to the impact parameter b presentation: f (b) = 1 2 d 2 q e
-iq b

dw1 = w1 · d d

X

dw2 n1 (w1 )n2 (w2 ) · w2 (w1 , w2 ), f ·

f (q ).

(9)

For the function under the module in eq. (8) we get (2) f (b) = 1 2 q M (-q 2 )ei · e-i 2 (E0 - q 2 ) q2 dq 2 2 M (-q 2 ) · J1 (q b) = (E0 - q ) u2 du 2 2 2 u + (E0 + qL ) b2 d2 q
q b

=

ni (wi ) =

2 ·d 2

d2 q

i

i

1 2 (qi )

2

=i = i b

2 w i m2 i 2 2 2 Wi,1 (i , qi ) + qi Wi,2 (i , qi ) , (3) Pi2

where Wi,1 and Wi,2 are the Lorentz scalar functions. All kinematic variables have the same definitions as in [3].

âM

-

x2 + u b2

2

J1 (u).

(10)


Yu.V. Kharlov and V.L. Korotkikh: Nuclear -radiation as a signature of ultra-peripheral ion collisions at the LHC 439

Here x = qL b = wb/A and u = q b. If we take Mel instead of the inelastic M as |Mel (-q 2 )|2 = Z2 2 F (-q 2 ) 4 el (11)

We take the inelastic form-factor from inelastic electron scattering off nuclei. A good parameterization of the inelastic form-factor is
2 2 22 F (q ) = 4 j (q R)e-q g 2

(21)

and put E0 = 0.0, then we get a well-known formula of the impact parameter-dependent equivalent photon number of the A2 nucleus (see (4) in [12]): N
(el) 2

(w, b) =

Z 2 1 · 2 b2
2

in Helm's model [13]. The squared transition radius is 2 equal to R = R2 + (2 + 3)g 2 , where R and g are the model parameters. According to (19) the reduced transition probability in this case is equal to B (E0 ) =
2 2 2 2 ZeR . 4

·

u2 du 2 J1 (u)Fel [-(x2 + u2 )/b2 ] , x + u2

(12)

(22)

For a point charge, Fel (q ) 1, we readily obtain N
(el) 2

So, the formulae for the process (1) are (13) d
A1 A2 A1 A2 X f

(w, b) =

Z 2 1 2 2 x K1 (x), 2 b2

=

in agreement with [3] at very large A . We write the form factors of the elastic and inelastic nuclear process in the same forms: F 2 (q ) =
2 0

dw1 w1 · d

X

dw2 () n (w1 )n2 (w2 ) · w2 1 (w1 , w2 ); (23) f
2 q 2 2 (E0 - qin ) 2

n

() 1

Z 2 (w1 ) = 2

d2 q

· (24) (25) (26) (27)

1 F 2 (q ) , 4 e2 Z 2
2

(14) -q 1,
q 0 2 2 in

1 F (q ) = 4 q

sin(q r)0 (r)rdr

(15)

2 · |F (-qin )|2 ; 2 2 w w E0 E0 2 = 2 +2 + 2 + q ; A A A

n2 (w2 ) = (16)
2 -qel =

Z 2 2 w A
2

d2 q

2 q 4 qel

2 2 Fel (-qel );

2 F (q ) = (2 + 1) 4

j (q r) (r, Z )r 2 dr
q 0

2 + q .



(4 )2 B (E ) q 2 , e Z 2 [(2 + 1)!!]2
2

(17)

where (r, Z ) is the nuclear transition density and B (E0 ) is the reduced transition probability . Then for the matrix elements M we get, in the limit q 0, |Mel (-q )| = |M (-q )| =
2 2 2 2

2 2 The value qin is close to qel at a large A factor at LHC energies. The effective two-photon luminosity can be expressed as 2

Z2 4 Z 4
2

F (q )
q 0

2 el

Z2 , 4

L(1 , 2 ) = 2 (18)
R
1

b1 db
() 1

1 R
2

b2 db

2 0

d · (2 , b2 )(B 2 ), (28)

·N

(1 , b1 )N

(el) 2

F (q )| Z2 4
2

2

q 0

(4 )2 B (E0 ) 2 q. e Z 2 [(2 + 1)!!]2

(19)

where R1 and R2 are the nuclear radii, (B 2 ) is the step function and B 2 = b2 + b2 - 2b1 b2 cos - (R1 + R2 )2 [3]. 1 2 Then the final cross-section is
A1 A2 A A2 X 1
f

The equivalent photon number for the inelastic process with A1 nuclear transition 0 will be N
() 1

Z 2 1 (w, b) = 2 2 b


d1 1

= d2 L(1 , 2 ) 2

X

f

(w1 , w2 ) .

(29)

â
0

u2 J1 (u)F [-(x2 + u2 )/b2 ] , du 2 in xin + u2
2 in

2

(20)
wE

2 Nuclear levels and form factors
The elastic form factor of a light nucleus is

as the generalization of (12). Here x
2 E0 2

= (E +

2 0

w2 2

+2

0

+ Fel (q 2 ) = exp -

)b .

2

r2 q 6

2

(30)


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The European Physical Journal A
1
1

10 10

-1 -2 -3 -4 -5 -6

1

F 2 (q )

F (q)

10 10 10 10

-3 -4 -5

2

2

10 10 10

2

-6

10 0 0.5 1 1. 5 2 2.5

0

0.2

0.4

0.6

0. 8

1

1.2

1.4

q, fm

-1

q, fm-1
Fig. 3. The elastic form factor (1) of 208 Pb and the inelastic form factor (2) of 208 Pb (3- , 2.615 MeV).

Fig. 2. The elastic form factor (1) of 16 O and the inelastic form factor (2) of 16 O (2+ , 6.92 MeV) from the electron scattering.

r2 = 2.73 fm for the nucleus 16 O. For a heavy with nucleus we take a modified Fermi nuclear density [14] 1 1 + -1 = (r) = 0 1 + exp -r-R 1 + exp r-R g g 0 sinh(R/g ) , cosh(R/g ) + cosh(r/g )
3

E0 = 2.615 MeV. This level is well studied experimentally [18] and has a large excitation cross-section. The reduced transition probability from the fit of the inelastic electron scattering on 208 Pb with excitation of the 3- level is [18] B (E0 3) = (6.12 105 ± 2.2%)e2 fm6 . We calculate the parameter 3 , using this B (E0 3), and take R and g from the density of the 208 Pb ground state: 3 = 0.113, R = 6.69 fm, g = 0.545 fm. Note that there are many levels higher than E0 = 2.615 MeV which decay to the first level of 208 Pb. This fact increases the event rate of the process (1), but we do not know the excitation cross-section of these levels. The elastic form factor (30) of 16 O and the inelastic form-factor of 16 O (2+ , 6.92 MeV) (21), corresponding to the electron scattering data, are shown in fig. 2. The same for a nucleus 208 Pb and the excited state 208 Pb (3- , 2.64 MeV) are shown in fig. 3. The squared inelastic form factor is less than the elastic form factor by more then two orders at small q < q0 (q0 = 0.5 fm-1 for 16 O and q0 = 0.4 fm-1 for 208 Pb). In the region of q > q0 they are comparable. The region of large q > q0 will contribute to the small impact parameter b. We are able to calculate the photon luminosity (28) for all regions of b to get the maximum electromagnetic cross-section of the process we are interested in. Then it should be possible to compare with experimental data in condition of clear selection of such process by the photon signal and the veto neutron or proton signal in the ZDC.

(31) (32)

0 =

3 4 R

1+

g R

2

-1

,

with the parameters for 208 Pb equal to R = 6.69 fm and g = 0.545 fm. This form of the density is close to the usual Fermi density at g R F (r) = 1
0 r -R g

(33)

1 + exp

and allows us to calculate the elastic form factor analytically: g sin(q R) coth( g q ) - cos(q R) . R (34) There are a few discrete levels of 16 O below the , p and n thresholds Eth () = 7.16 MeV, Eth (p) = 12.1 MeV, Eth (n) = 15.7 MeV [15]. The level 2+ at E0 = 6.92 MeV is the strongest excited one in the electron scattering. The parameters from the inelastic electron scattering fit on 16 O with excitation of 2+ level (E0 = 6.92 MeV) are [16] Fel (q ) = 2 = 0.30, R = 2.98 fm, g = 0.93 fm. They correspond to B (E0 2) = (36.1 ± 3.4)e2 fm4 . (35) 4 2 Rg 0 q sinh( g q )

3 Angular and energy distributions of secondary nuclear photons
We suppose that the nucleus A (µ) in process (1) is un1 polarized. At this point now we do not know the relative excitation probability of |µ states, where µ is a pro jection of spin . This assumption needs further study in the

There are more than 70 discrete levels of 208 Pb [17] below the neutron threshold Eth (n) = 7.367 MeV. About 30% of the levels decay to the first 3- level of 208 Pb at


Yu.V. Kharlov and V.L. Korotkikh: Nuclear -radiation as a signature of ultra-peripheral ion collisions at the LHC 441
60 50 40

30

20

10 9 8 7 6 0 100 200 300 400 500

Fig. 4. Nuclear photon energy as function of its polar angle in the laboratory system at LHC energies for two nuclei: 16 O (2+ 0+ , 6.92 MeV) (1) and 208 Pb (3- 0+ , 2.615 MeV) (2). ZDC marks the region of the zero-degree calorimeter in the CMS.

Fig. 5. Transverse ZDC plane. The points are the simulated hits of neutrons (top) and photons (bottom) from ref. [21].

future. So we use a formula (27) in our work [10] for the angular distribution of secondary photons, which is valid for isotropic photon distribution in the rest system of A 1 according to equal probabilities of excitation. If we calculate the integral cross-section of reaction (1) using eq. (29), then the angular and energy distribution of photons are equal to [10] dA = d dA = dE
A1 A2 A A2 X
1

· ·

2 (1 +
2 A
1

2 A

1

sin



tan )2 · cos3

2

, (36) (37)

A1 A2 A A2 X 1

(2A1 E0 - E ) , 2A E0 1

where (x) is the step function. The angular distribution does not depend on the photon energy and the energy distribution is uniform. The photon energy E and the polar angle in the laboratory system are defined as E =
A
1

E0 (1 + cos )
1

on is shown in fig. 4. Thus the energy E will depend on the position of photon hit. Our calculations with the TPHIC event generator [19] show that a deflection of the direction pA from pbeam at LHC energies in the reaction (1) is very small at large A , 0.5 µrad. In the experiments CMS and ALICE, which are planned at LHC (CERN), the zero-degree calorimeter [20, 21] was suggested for the registration of nuclear neutrons after ion interaction. We demonstrate a schematic figure of the ZDC CMS at a distance L = 140 m in the plane transverse to the beam direction in fig. 5. The CMS group also plans to include the electromagnetic calorimeter in front of the ZDC. As an example, we show the angular distributions (36) in arbitrary units and the energy dependence (38) on the (x, y ) coordinates of the ZDC CMS for the two nuclei 16 O and 208 Pb in fig. 6. The direction of the nucleus A1 coincides here with the beam direction. The point (x, y ) = (0, 0) is the center of the ZDC plane.

= 2 tan =

A

E0 /(1 +

2 A

1

tan2 ),

(38) (39)

1
A
1

sin , 1 + cos

4 Cross-section of the pro cess with the nuclear radiation
We demonstrate our results for the c (2.979) production. The previous results [3] used old values of the widths and a point nuclear charge. Now we take resonance parameters

where and are the polar angles of the nuclear photon in the rest nuclear system and in the laboratory system with an axis z||pA . The photon energy E dependence


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The European Physical Journal A

O

208

Pb

3500 3000 2500 2000 1500 1000 500 0 4 3

2500 2000 1500 1000 500 0 4 2 1 5 0 34 -1 12 -2 -1 0 -3 -2 -4 -5 -4 -3 x 3 2 1

Number

y

y

0 45 -1 23 -2 01 -3 -2 -1 -4 -5 -4 -3 x

16

O

208

Pb

50 40

14 12 10 8 6 4 2 3 0 4 2 1 0 45 -1 23 -2 01 -3 -2 -1 -4 -5 -4 -3 x 3

E, GeV

30 20 10 0 4

2

1

y

y

0 -1 -2 -3 -4

45 23 01 -1 -3 -2 x -5 -4

Fig. 6. The photon angular distributions (upper row) and the energy dependence (lower row) for 16 O (2+ , 6.92 MeV) (left column) and 208 Pb (3- , 2.62 MeV) (right column) radiation decay in the laboratory system on the ZDC plane (x, y ) at 140 m distance from point interaction. x, (cm) is the horizontal and y , (cm) is the vertical axis. The photon energy interval in the ZDC region is 19­48 GeV for 16 O (2+ ) and 7­14 GeV for 208 Pb (3- ).

from the review of particle physics [22], c = 4.8 keV, and a realistic charge distribution. The calculations was made with the help of TPHIC event generator [19]. We use a well-known formula [2] of the narrow resonance cross-section:
2 (W 2 - MX )/MX , (40) where W 2 = 4w1 w2 , X and MX is the spin and mass of the resonance. The LHC luminosity and our results according to (29) and (28) are in table 1 for the process (1) with Afinal = A1 or A . 1 Our results in table 1 show that though the crosssection of the process (1) for the nucleus 208 Pb is larger than that for 16 O, the event rate is smaller because of the



X

(w1 , w2 ) = 8 2 (2X + 1)

X

lower LHC luminosity for 208 Pb. The cross-section with a nuclear excitation is smaller by three orders of magnitude than that without the excitation, since the intensity of excitation is not large and the inelastic form factor is smaller than the elastic form factor (see figs. 2 and 3). Therefore for the accepted LHC luminosities it is possible to use secondary photons as a signature of clear electromagnetic nuclear processes only for the production Xf with rather large cross-section X . Light ions are more preferable than heavy ions to detect the nuclear radiation.

5 Conclusion
In this work we suggest a new signature of the peripheral ion collisions.


Yu.V. Kharlov and V.L. Korotkikh: Nuclear -radiation as a signature of ultra-peripheral ion collisions at the LHC 443 Table 1. Cross-section of c (2.979 GeV) production by fusion. A
final

L (cm

-2

s

-1

)

L (pb

-1

)

356 µb 73 nb 296 µb 129 nb 66 nb 0.201 nb

event/106 s 147000 1020000 122000 53 926000 2810

Point charge of the nuclei
208 16

Pb82 O8 Pb82 Pb (3- ) 82

4.2· 10 1.4· 10 4.2· 10 4.2· 10 1.4· 10 1.4· 10

26 31

0.00042 14.0 0.00042 0.00042 14.0 14.0

With form factors of the nucleus and in the region R < b <
208 208 16 16 26 26 31 31

O8 O (2+ ) 8

The formalism of the process (1) is developed in the frame of the equivalent photon approximation. The new point is the introduction of the inelastic nuclear form factor. It allows to consider the excitation of discrete nuclear levels and their following radiation decay. It is shown that the energy of this secondary photons are in the GeV region due to a large Lorentz boost at LHC energies. The angular distribution of the photons has a peculiar form as a function of polar angle in the beam direction. The ma jority of photons fly in the region of angles of a few hundred micro-radians, which are those detactable in the ZDC CMS and ALICE experiments. Thus the nuclear radiation is a good signature of clear peripheral ion collisions at LHC energies when A and Z of the beam ion are conserved. The trigger requirements will include a signal in the central rapidity region of particles from Xf decay, a signal of photons in the electromagnetic detector in front of the zero-degree calorimeter and a veto signal of neutrons in the ZDC. We suggest to use the veto signal of the neutron in order to avoid the processes with nuclear decay into nucleon fragments. The nuclear radiation can be used for tagging the events with particle production in the central rapidity region in ultra-peripheral collisions. Light nuclei are more preferable in comparison with heavy ions, since they have higher beam luminosity at LHC. The cross-sections of the process with the nuclear excitation are three orders of magnitude smaller than the one without excitation. The accepted nuclear luminosities enable us to use this signature for the large cross-section of the Xf system production.
Authors are very grateful to L.I. Sarycheva and S.A. Sadovsky for the useful discussions and K. Hencken and R. Vogt for helpful comments.

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