Fortschritte der Physik / Progress of Physics: Volume 32, Number 4 [Reprint 2022 ed.] 9783112656129

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 32 • 1984 Number 4

Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J . Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay

CONTENTS: A . OKADA, K . M I T S U I , T . KITAMXTRA, e t a l .

Inelastic Scattering of Cosmic Ray Muons on Iron Nuclei and the Virtual Photon Shadowing

135—173

B . K . BANDYOPADHYAY, a n d A . N . S E N G U P T A

Magnetic Monopoles — A Brief Review G. MARX

The Thermodynamical History of the Universe

175—184 185—206

AKADEMIE-VERLAG • BERLIN ISSN 0015 - 8208

Fortschr. Phys., Berlin 82 (1984) 4, 135-206

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from " 1 " onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary uprigth typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vektors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp, ...): black underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upringth Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c G, k K, o O, p P, s 8, u U, v V, w W, x X, y Y, z Z). I t will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, b\, Mil, Mij, Wnf Please differentiate between following symbols: a, « ; «, a , oo; a, d; c, 0,

160 180 200

Photon energy (GeV)

Fig. 2. Photon-proton cross section vs. photon energy (from Review of Particle Properties)

Most of the phenomenological formulas are based on some assumptions either on the form of i\ or on that of a T . One way is to use approximately [12] F2**Q*fm

(4)

where f(Q2) is a certain funftion, and the other way is to express [26]

{o^Twia-*Av)

{w = 1,2)

(5)

where 50 GeV), though Callan-Gross relation may have no physical meaning at such low Q2. The combination of eq. (5) and R = 0 leads to a very simple expression of dajdy as an integration of eq. (2) by Q2 [22, 30], being

10

01

:001

0.2

Fig. 3

OA

0 6

08

1.0

Fig. 4

Fig. 3. y (= v/E) dependence of y • dojdy for empirical expressions of inelastic leptonproton interaction with an fixed v 50 GeV. All formulas in Appendix A including those not drawn here have nearly the same cross section at y = 0.55. Beyond there, each cross section is in the following order; BRASSE R = BEZRUKOV

(fjv

1

without shadowing

BRASSE R

=

0.18

KIRK

R

=

0 . 5 2 (_•

(—) )

KIRK

R

=

1.2(1 -

x)/Q2

(

).

z

Fig. 4 . y dependence of the median Q for B R A S S E ' S expression with R = Q 2 / V 2 . v is fixed to 50 GeV. The median Q2 increases very slowly with increasing v.

141

Fortschr. Phys. 82 (1984) 4

given by da T~ dy

=

x I f a Y — n rN y \y f

—

EH 1 -

X In

1 -

1 y)

M'

\

\

o

I.

m

m0 \

m 0 2 (l — y)J

Ü

2M

nit

(8)

where, y = v/E. But we have to keep in mind that there are another expectations of R by QCD [31] etc. Several dajdy distributions (y = v/E) from the formulas in Appendix A are shown in Fig. 3 at v = 50 GeV. da/dy is almost independent of v, when v is larger than ~ 50 GeV, for any formula except for Bezrukov's which takes into account the rising cross section of ayN like shown in Fig. 2. The differences of slope in da/dy are mainly due to the different assumptions of R among the formulas. To explain this, we show in Fig. 4 the median Qz value vs. y, and in Fig. 5 the representative value of e (see eq. (2))'for the median Q2. Here both were calculated by Brasse's formula with R = Q2/v2. It is seen that the relative contribution of aL to the muon cross section becomes smaller at higher y. So the assumption of R = 0 gives the most gentle slope. Concerning v dependence of the cross section mentioned above, it seems natural to expect that the cross section of the virtual photon at low Q2 increases up to a considerable high energy, just like Bezrukov's formula, in view of both the GVD success and the experimental fact of the rising cross sections of hadrons. 100 \electron-pair

10

-

bremsStrahlung

E XI

4

10 08

1

V

knock-on

\\ \\ \ \

5>>

06

N.-

\

01

04

\ \ \ \ \ \

knock-on

02

02 Fig. 5

04

y

06

0.8

1.0

001

t 01

i 10

Fig. 6

Fig. 5. The representative value of the polarization factor corresponding to the median Q2 in Fig. 4 vs. y. Fig. 6. y dependence of y • dajdy per nucleon in Fe for the three electromagnetic processes in Appendix B, under fixing v as, v = 50 GeV (solid curve), and v = 1000 GeV (broken curve).

142

A. OKADA et al., Inelastic Scattering of Cosmic Ray Muons

Fig. 7. The expected event ratio of hadronic showers to electromagnetic ones for the empirical expressions in Appendix A, taking into account the muon spectrum at the zenith angle of 88° from eq. (10) in text.

The target in the present experiment is iron nucleus. So f a r we have considered only a. nucleon as the target and have taken it for granted t h a t there is no difference between proton" and neutron target. I n Fig. 8 [32] is shown the experimental difference of cross section between them. 2 ) We could ignore the difference a t very small x' = Q2j/(2Mv M2) 2 3 and small Q (see 4° d a t a in the figure), ) where the diffractive process is expected to dominate. On the contrary, nuclear shadowing of photons is not negligible. If the hadronic component of the physical photon is present over a long range (i.e. if tf in eq. (3) is much longer t h a n the nuclear radius), then interactions deep inside the nucleus will tend t o be shadowed by those near the surface on the incoming side. The total cross section on 1.61

1

1

1

1

1

12b

i i 1 1— A 6"-10" Dato D 18"-34" Data x 4" Data o 15"-34"Data. • 50", 60" Dato

as H " Ì3 £ •o

.zia

0.4 • 0 25 02

_i

I OA

I

L 0.6

0.8

10

x' Fig. 8. The ratio of the neutron cross section to proton cross section (from Ref. [32]) 2 3

) See also Ref. [37]. ) For example, MAY et al. [41] used (CRB/CRP) = 1 — x' [33].

143

Fortschr. Phys. 32 (1984) 4

the nucleus of mass number A is expected o{A)=

Atlp(l)

< Aa{\)

(9)

where a( 1) is a suitable weighted average of total cross section on neutrons and protons. The strength of shadowing is usually represented by AtU/A or n defined as A1' = Atif. Experimental data of real photons exhibit the shadowing effect as seen in Fig. 9 [34], which suggests more shadowing at higher energy. Fig. 10 [35] is an example of the virtual photon shadowing vs. Q2 with energy ranging from 40 to 200 GeV. I t seems that the shadowing of virtual photon in the range of Q2 < 1 GeV2 is comparable to that of real photon for Er > 50 GeV. However, at lower energies (v < 10 GeV), most of experiments claim a much narrower Q2 range (Q2 < 0 . 1 GeV2) [16, 36—39] for the significant shadowing. 4 ) Moreover in the case of heavy nucleus targets (A > 100), some experi{C 0.8-

< —
plus non-shadowing 'point-like' picture), diagonal GVD [44], off-diagonal GVD [45], 'self-absorption' feature [46], etc. There is, however, no definite theoretical work to explain, at the same time, the increasing of shadowing still at such high energies a s i > ~ 100 GeV both for Atii/A and for Q2 range 5 ) as seen in relatively new data, i.e. Fig. 9 and Fig. 10 6 ). From this point of view, it is interesting to investigate experimentally the shadowing behaviour further at still higher energies.

3.

Apparatus and Operation

Detailed descriptions of the spectrometer and the calorimeter have been given elsewhere [1, 2, 4]. A schematic view of the whole arrangement of MIJTRON is shown in Fig. 11, and a more detailed view focussed on the calorimeter only is in Fig. 12. The calorimeter consists of 9 layers of proportional chambers, 6 layers of optical spark "chambers and 10 layers of iron plates with a total depth of 1130 g/cm2. Above the calorimeter 3 m distant) are there 6 channels of scintillation counters with size of 50 X 50 cm 2 each to detect local air shower backgrounds. Magnetic spectrometers with the maximum detectable momentum (MDM) of 3 TeV/c on the average are arranged both upstream and downstream of the calorimeter. For muons traversing throughout the both spectrometers without any large energy deposition, the MDM is estimated to be 22 TeV/c, 1

9 1

3

2

TOF

N Magnet 1m

A

wsc

5

7

6

Calorimeter

10 3

proportional counter

2

S Magnet

Schematic view of MUTRON

8 U

TOF

(side view)

Fig. 11. Schematic view of M U T R O N (side view). Wire spark chambers (No. 1 — 10) are used for the determination of muon trajectory. No. 9 and 10 were added after three years operation. Proportional counters (No. 1 — 4 ) with 2.4 cm read-out spacing are used for the triggering. The direction of the magnetic field in the iron magnet is perpendicular to the figure. 6)

Using their data, the authors of ref. [37] show that Aett/A can be described as a function of scaling variable x' = Q2j{2Mv + Mz). This scaling behavior looks natural from eq. (3) and gives us an intuitive comprehension of the extended Q2 range a t high energies seen in Fig. 10. B u t , of course, strict scaling conflicts with real photon data (i.e. x = 0) which varies with energy like Fig. 9. 6 ) The authors of ref. [35] conclude that, in the context of GVD, their data (Fig. 10) provides evidence that higher mass vector mesons than the p meson participate in shadowing at these high energies.

Fortschr. Phys. 32 (1984) 4

145

which is used to study the momentum spectrum of cosmic ray muons. Track detectors in the spectrometers are wire spark chambers with magnetostrictive readout. Space resolution of the wire spark chamber including the alignment error is 0.8 mm expressed as the standard deviation and the detection efficiency of muon track in a wire spark chamber is 90%, both in a sense of the average. Optimal spark chambers in the calorimeter were operated only for a part 1/10) of the running time to search for anomalous events and to check the system of the calorimeter. We do not touch this subject hereafter in the present study. a) top view

Fig. 12. Schematic view of the calorimeter. Each layer of proportional chambers is composed of four independent chambers as seen in top view. The iron plate is 12 cm thick.

MUTRON has two trigger modes that follow: 1) spectrometer trigger mode The whole system is triggered for the events which satisfy the condition that the deflection angle in the magnets determined by P R 1, PR2, P R 3 and P R 4 in Pig. 11 is smaller than a certain value [1], corresponding momentum of muon being about > 80GeV/c. The geometrical acceptance is 0.12 m 2 sr, being nearly constant above 100 GeV/c with the range of zenith angle from 86°—90°. This trigger mode was adopted to study the muon spectrum at relatively high momenta. Events accompanied by rather large cascade showers in the calorimeter are used for the present study. Though the geo-

146

A. O K A D A et al., Inelastic Scattering of Cosmic Ray Muons

metrical acceptance is small compared to the calorimeter trigger mode below, this trigger mode has an important advantage that the trigger efficiency is independent of those longitudinal shapes of cascade showers, which are used for the identification of hadronic showers. Moreover, the fact that both momenta of incoming and outgoing muon can be measured for almost all events obtained by this mode gives a way to the energy calibration of the calorimeter as described in Section 6. 2) calorimeter (shower) trigger mode The whole system is triggered for events which satisfy both of the following conditions; a) each signal from any three successive layers is greater than or equal to 16 equivalent particles, among the proportional chambers from the second to 8th layers, b) each signal from the first and 9th layers is greater than or equal to single particle including the fluctuation. The geometrical acceptance is 6.0 m2sr with the zenith angle ranging from 45° to 90°. For the geometrical reason, neither momentum of incoming and outgoing muon is measureable in about 90% of the events. However, even for such events, wire spark chambers near the calorimeter inform us of the muon direction and the existence of showers from outside which of course are rejected from the present study. These two trigger modes have been operated in parallel throughout almost all running time. The total net running time is 7.14- 105 min for the spectrometer trigger and 6.44 • 105 min for the calorimeter trigger. Each observed event has a flag to indicate the calorimeter trigger, but we failed to set another flag about the spectrometer trigger. So when the flag of the calorimeter trigger shows 'y e s ' ™ a n event, no one can judge whether the spectrometer trigger is also 'yes' at the same time or 'no'. This situation forces us to a somewhat complicated treatment of data as will be seen later on. Here, for the convenience of data analysis, we mention a little more about the geometrical acceptance. Calculated acceptances are shown in Table 2 in each case of three trigger conditions; (1) calorimeter trigger, (2) spectrometer trigger, and (3) the both simultaneous trigger. Further each case is devided into three categories according to the number of measureable muon momenta in an event, namely (1) both incoming and outgoing momentum (P, n n P out ), (2) incoming and/or outgoing momentum (P in u Pout); and (3) no restriction (P u P). These acceptances include weights due to the angular dependences of the muon flux and the target (iron plates) depth, and are normalized to the muon flux at 88° (near the average angle for the spectrum study) and to the target depth of the case that the direction of muon is normal to the iron plates. In fact, thus defined acceptances vary with muon energy. The values in Table 2 are for ~ 5 TeV. The difference arising from this variation has to be included in the actual analysis. The muon flux used in the above calculation and also during data analysis following is an approximated expression [47]: U(E, 6) = A

+ | IK„ j • W(Xl),

(10)

where and IKli are muon intensities produced from pion and kaons, respectively, and are approximated like

Fortschr. Phys. 32 (1984) 4

147

IKv, is obtained by substituting K for every suffix n. The survival probability W(xi) is given by 1.08 GeV

W{xJ

=

«cos e' + (a+bE)-x„ x0

(cos 6' = 0.75 cos 0 + 0.25 cos 6*) with the energy at production, = E + {a - f bE) (x0 - xj/cos

0',

and the production depth ; x x = 100 g/cm2 • cos 0*. The zenith angle 0* at production is represented by 10* =

^E

• a

.Re + h(x1)

where /¿E is the earth radius and h{x) is an approximation of the height at the atmospheric depth x for US standard atmosphere [48]. The constants are, BN = 1 1 6 GeV, BK = 855 GeV, r„ = mh/mn, rK =

m^m^,

x0 = 1033 g/cm2, a = 2.5 • 10^3 GeV/(g/cm2), b = 2.78 - 10" 6 (g/cm 2 ) 1 , and y = 2.7. The value of y different from 2.585 of the reference [47] comes from the best fit of MUTRON data [4] to the above expression at the zenith angle of 88.8° on the average. Table 2 Geometrical acceptances (m2 sr) at an enough high momentum of the incident muon (P l n = 5 TeV/c)

4.

trigger mode

Category 1

Category 2

Category 3

spectrometer trigger calorimeter trigger the both simultaneous trigger

0.104 0.175 0.090

0.108 0.417 0.094

0.113 3.850 0.098

Trigger Efficiency and Monte Carlo Simulation

Trigger efficiencies were estimated by Monte Carlo methods. In case of the spectrometer trigger mode, the efficiency is governed mainly by the trigger logic mentioned in Section 3, and so depends on the incident and outgoing muon momenta. We take into account not only the bending trajectory of muon in the magnets, but also multiple Coulomb scattering and constant energy loss in them. After simulating such trajectories considering the geometrical weight, we get the probability that they satisfy the trigger logic. Supposing the four-fold detection efficiency of PR1 ~ P R 4 in Fig. 11 is 95% [1] besides the trigger logic, which gives no significant effect to the final result, the trigger efficiency for the spectrometer trigger mode is obtained as shown in Fig. 15.

148

A.

OKADA

et al., Inelastic Scattering of Cosmic Ray Muons

In case of the calorimeter trigger mode, the trigger efficiency depends on the longitudinal structure of the individual shower because of the condition (a) described in Section 3. Naturally it also depends on the sampling depth, or the angle of shower axis with respect to the iron plates in the calorimeter. A Monte Carlo simulation of the behaviour of muon interactions in the calorimeter was carried out to estimate trigger efficiencies for the electromagnetic showers and hadronic showers, and also to estimate the other quantities in the following Sections.

1000-

"

%

o

o O a-100

T>

«3 ^

o

•3

% O o •o o • o

10

10

20

30

40

cu

50

60

70

K g . 13. Comparison between an experiment from Ref. [51]) (•) and the present Monte Carlo simulation ( o ) for the hadronic cascade curve by an incident 200 GeV proton.

All interactions are simulated one-dimensionally by a full Monte Carlo method. Simulations of electromagnetic cascade showers initiated from electrons or gamma rays which are produced through electromagnetic interactions of muon as well as through decays of neutral pions during hadronic cascade developments are based on the approximation B [72]. Simulations of hadronic cascade processes are similar to W. V. JONES' [49], which reproduces the recent accelerator data [51] pretty well (e.g. Fig. 13) in spite of rather old-fashioned interaction model, i.e. CKP [50]. As for the muon-nucleus interaction as the first stage of a hadronic cascade, we assume that the characteristics of the secondary particles are the same as those in nucleon-nucleus interaction except for the absence of leading nucleon.7) Species of the secondary particles are assumed to be limitted to only pion, and the interaction length of charged pion is 11.3 r.l. in Fe. Some fraction of energy is lost in each interaction owing to the evaporation of the target nucleus just like JONES'. The transition curves for simulated electromagnetic and hadronic showers in Fe are shown in Fig. 14. We have to take into account accompanying showers besides a main showers, produced through succesive interactions of a muon while it traverses the calorimeter. These interactions are bremsstrahlung [53] direct electron-pair production [55] and knock-on electron production [56] (see Appendix B). Finally we have to consider the resolution of ") As for hadrons produced in muon-proton interaction, see Ref. [52].

Fortschr. Phys. 32 (1984) 4

149

the shower detector, proportional chamber including its readout circuits, which is approximated by [5]: AN/N = 5.5 /N + 1 . 5 / l / F + 0.1, where N is the number of shower particles equivalent to the pulse height of signal.

Fig. 14. Average transition curves for electromagnetic (solid curves) and hadronic (dashed curves) cascade showers by the present Monte Carlo simulation.

Calculated trigger efficiencies are shown in Fig. 16 for each category mentioned in Section 3, that is 'Category 1' [P i n n Pout]> 'Category 2' [P i n u P o u t ] and 'Category 3' [ P u P ] . Since the range of acceptable zenith angle is different among the three categories, and then the averaged sampling depth is different, the resultant trigger efficieny is different. The trigger efficiency for the hadronic shower is higher than that for the 1000

P(n (GeV/c)

Fig. 15. The trigger efficiency for 'spectrometer trigger' vs. outgoing muon momentum with different incident muon momenta.

100

1000 P out (GeV/c)

150

A. OKADA et al., Inelastic Scattering of Cosmic Ray Muons 10r

-1000 500 » (GeV) 200 100

* 0.1L_ a cri cn (a> 001

0.1

y

10

Fig. 16(a): electromagnetic showers for 'category 1' = 'category 2' (approximated)

2000 1000 500 v (GeV) 200

Fig. 16 (b): electromagnetic showers for 'category 3'

Fig. 16 (c): hadronic showers for 'category 1, = 'category 2' (solid curves) and'category 3, (dashed curve)

The trigger efficiency for 'calorimeter trigger' vs. y (= v/E) for different v's.

electromagnetic shower because of its longer tail. The general tendency that the trigger efficiency increases at low y region is due to the accompanying showers described just above.

5.

Separation of Hadronic Showers

T h e least square f i t to the theoretical electromagnetic shower curve with t w o parameters of shower energy (e) and starting point (i 0 ) of the shower is applied to each event [22], using signals f r o m three successive layers, requiring that the highest signal in an event is included among the three. Since the biggest electromagnetic shower in an event comes usually f r o m muon bremsstrahlung, the theoretical curve to be fitted must correspond to that of incident

151

Fortschr. Phys. 3 2 (1984) 4

two-electron. I t is given by i(e,

t) =

g(e,

t)

+

dg(e,t)

9

7

(12)

dt

using approximated expression of incident one-photon [57] : ,

> '

/«

0-31 f ¥

=

e

x

3

3311

,

X

(13)

p

where t denotes depth in radiation length and Y = In (e/e0) with critical energy s 0 («a 20.7 MeV for Fe). Then the chi-square for the fit is written like ;+2 *

where,

=

2

27 \i=j

[».

^

U -

to)]2

M

(14)

minimum

: equivalent number of particles in the i-th layer, depth (r.l.) of the i-th layer, and r-2

= [0.2w(e, ti -

i0)]2 +

h -

t0) +

(15)

100.

The expression of a t 2 above was chosen so as to keep the mean of each term in eq. (14) nearly constant against ni and i, for the simulated electromagnetic events.

0

log (5)

Fig. 17. f distribution (see t e x t ) . 'Hadron-like' events correspond to Log (f) 0. 'Electron-like' events correspond to Log (f) < 0. Histogram by solid line ; data e > 200 GeV Histogram by dashed line ; data E > 500 GeV Solid curve ; Monte 'Carlo simulation E = 200 GeV Dashed curve; Monte Carlo simulation e = 5 0 0 GeV' 2

Fortschr. Phys., Bd. 32, H. 4

10

152

A.

OKADA

et al., Inelastic Scattering of Cosmic Ray Muons

Now let us introduce another quantity defined as 3+4

v = Zini

i=/+3

/3+2

— n(E
50 GeV) Fig. 21 (b): 'hadron-like' events (e ^ 50 GeV)

fraction of invisible energy in the detector is greater for the hadronics hower than for the electromagnetic shower because of nuclear evaporation, neutron escape, etc. In the Monte Carlo simulation, the ratio of the visible energy of the hadronic shower to that of the electromagnetic shower is taken to be 80% arount 50 GeV and ~ 85% at very high energy. The Monte Carlo simulation fairly reproduces our data, i.e. the asymmetric distribution in Fig. 21(b). As this effect of invisible energy is of considerable importance considering the steepness of muon momentum spectrum, we will take it into account in the data analysis in section 7, as well as the escape of the hadronic shower tail out of the calorimeter, which is estimated to be about 5.3% of e on the average. Accelerator data for the visible energy ratio of the hadronic to the electromagnetic shower [58] are given in Fig. 22. The direct comparison with the present case may be not

156

A.

OKADA

et al., Inelastic Scattering of Cosmic Ray Muons

so fruitfull by reason of the differences of the detector and the sampling thickness. We have to allow this problem to remain a little ambiguous for the present study. Concerning the energy resolution, Fig. 21 also confirms the validity of the Monte Carlo simulation. Although the hadronlike events might give a somewhat worse resolution and moreover the energy dependence of the resolution is hardly confirmed because E»,jiWe (pions)/ Evij,ue (electrons) c

10

missing in hadronic shower t * o 8» * B

°CO

+

*+

•i

05

• o this experiment (K^g/cm1)

0

J 20

1 40

l 60

1 80

I 100

1 120

L 140

Ett.JGeV)

Fig. 22. An example of experimental data about the visible energy ratio of the hadronic to the electromagnetic shower (from Ref. [58])

of insufficient statistics, we will use those resolutions obtained by the Monte Carlo simulation, examples of which are given in Fig. 23. The dip seen in the curve for the electromagnetic showers in Fig. 23(a) is due to the imperiodic structure of the calorimeter. In our data analysis in the following section, the curve is approximated by a weighted sum of two different Gaussians. For the hadronic showers, the corresponding curve is approximated by one Gaussian. The energy resolutions in terms of the standard

Fig. 23. The distribution of shower energy to be detected (E), expressed as the fraction to a fixed true shower energy (v), from the present Monte Carlo simulation without the consideration of trigger efficiency. Fig. 23 (a): electromagnetic shower at v = 50 GeV (solid curve) and v = 1000 GeV (dotted curve). A dip in the curve is due to the imperiodic structure of the calorimeter. We approximated the curve by the sum of two Gaussians. Fig. 23 (b): hadronic shower at v = 50 GeV (solid curve) and v = 1000 GeV (dotted curve).

157

Fortschr. Phys. 32 (1984) 4

deviation are expressed by, Ae/s ~ 9.6 + 2.45/|/e(TeV) % for t h e hadronio showers a n d Ae/e ~ 23.1 + 3.25/j/e(TeV) % for t h e electrogenetic showers. T h e choise of t h e energy resolutions are n o t so i m p o r t a n t for t h e final results. T h e deposited energy in t h e calorimeter h a v i n g been discussed includes t h e contrib u t i o n f r o m t h e additional showers described in Section 4 a n d 5. T h e s u b t r a c t i o n of t h i s c o n t r i b u t i o n is necessary for us t o e s t i m a t e t h e energy of t h e m a i n shower of interest. I n Fig. 24 we show this c o n t r i b u t i o n in t e r m s of t h e average ratio of t h e whole deposited e n e r g y s f r o m eq. (18) t o t h e t r u e m a i n shower energy. T h e ratio d e p e n d s on y, a n d in case of t h e calorimeter trigger mode, it also depends on t h e shower energy, because additional showers a n d actually observed e correlate t o t h e trigger efficiency more s t r o n g l y a t lower energy. 1 2t

u o cn a

1.1

< 10 001

01

y

10

Kg. 24. The contribution of the additionally accompanying showers in the calorimeter, expressed by the ratio of the whole energy to be detected to the main shower energy from the present Monte Carlo simulation.

The correction arising f r o m t h e additional showers m a y be called correction for ' e x t e r n a l radiation'. On t h e other h a n d , a n o t h e r correction for ' i n t e r n a l radiation' seems also effective to t h e estimation of t h e t r u e transferred energy a t t h e inelastic interaction. T h e i n t e r n a l radiation m e a n s t h e emission of real p h o t o n s f r o m m u o n during t h e inelastic interaction. T h e probability of p h o t o n emissions w i t h t o t a l p h o t o n energy between Ey a n d Ey dKy is given b y [59] (using a Bloch-Nordsieck t y p e of a p p r o x i m a t i o n ) : dP = ccF(y, ) and Q2 is given by = P 3 (2 + g,) , ( 1 - * ) » » + - P 5 £ ••

(1 - xr*

^

^

where 9s = 9oa + e. 0s = 005 + e, e = x In [{Q2 + m 0 2 )/w 0 2 ], and Ps, P5, g03, g05, x and m0 are free parameters. The first term considers scattering from a state of three valence quarks and g3 gluons. The second term considers scattering from a state which has a quark-antiquark pair in the sea, and includes an explicit Q2 dependence for the approach to scaling in the GVD spirit. For the fit, the combined Fermilab data [16], MIT-SLAC data [68] and SLAC data [68] were used. Two expressions for R were considered R = 0.52,

and

R = 1.2(1 - x)/Q2.

Fitted parameters for each R are in Table A-2 together with ay P = +

where m^ = 0.54 GeV2, m 2 2 = 1.8 GeV2 and £ = 0.25. The function G(z) which takes into account the shadowing is defined as G(z) = A | J -

1 + e - ( l + z)j,

z = 0.00282 A"%rP(v).

In case of the nucleon target, G(z) is substituted by G(z) = 1. 3*

170

A.

OKADA

et al., Inelastic Scattering of Cosmic Ray Muons

The above expressions for virtual photon cross sections are based on the nc ndiagonal GVD model, and in agreement with Fermilab data [16] and SLAC data [68,, if cryP is approximated by, aYp(„) =

114.3 +

1 . 6 4 7 I n 2 ( 0 . 0 2 1 3 V ) [xb

where v is in unit of GeV. The expression of