arXiv:hep-ph/0411081v1 5 Nov 2004 etuo n/rCiosa vprtn iibakholes black Tomaras N. mini Theodore evaporating as Chirons and/or Centauros 02,adb rn ..Euain-Hrkio from Heraklitos - Education. Education of P.O. Ministry grant Hellenic a HPRN-CT-2000- the contract by RTN and the under 00122, EU the by part ∗ a ..o 28 10 ealo,Cee Hellas Fo.R.T.H. Crete, and Heraklion, 71003 2208, P.O.Box na1 iesoa ol (the world dimensional 10 a in esrn hoy sta u pctm sa4di- 4 a (a is hypersurface spacetime our mensional that Su- is in Theory, dimensions accomodated perstring extra easily assumption, large basic The with gravity TeV-scale Introduction 1. Centau the of i ones produced the possibly to similar and quite dimensions, characteristics extra have large with models inlsaet eof be gravita- to fundamental scale a tional for the grav- allows that the force, via fact is itational The bulk the bulk. with communication the grav- only Only in propagate subspace. can dimensional con- ity 4 are our gluons) on Z, of fined W, Model (photon, Standard the fun- of the of interactions carriers damental the as well as (, higgses), matter leptons, known all that feature additional ino iibakhls(Bs,peitdin predicted (MBHs), holes evapora- black subsequent mini and of formation tion the be to may (CLEs) due events Centauro-like known long exper- ac- cosmic-ray forthcoming iments. and the neutrino in (LHC), years celerator few effects, next gravity the quantum within large and physics, dimensions string higher observing of prospect mm. citing a of fraction a may as dimensions large spatial as extra the be of size the time hstl smsl ae n[] eerhspotdin supported Research [1]. on based mostly is talk This eateto hsc n nttt fPam hsc,University Physics, Plasma of Institute and Physics of Department ti rudta h inl xetdfo h vprto o evaporation the from expected signals the that argued is It h hoeia rmwr fti aki the is talk this of framework theoretical The vnbte,i hstl ilageta the that argue will I talk this in better, Even ex- the to leads true, if scenario, a such Clearly, O a 1e) hl ttesame the at while (1TeV), ∗ D3-brane bulk embedded ) ,wt the with ), . 1 oeet,a l ytr fcsi-a physics. cosmic-ray of mystery old an events, ro ∇ equation Poisson responding point a ilΦ( tial radius of with circle space a R dimension, Minkowski transverse dimensional extra 4 one a with ing ihrdmninltorus R forming dimensional and higher compact a dimensions would extra which con- the to brane, sider and background, the gravitational the on to modify tension is scenario the straightforward theoretical neglect most above the the of excluded, realization priori a not models gravity TeV-scale [3]. 2. and [2] in described are picture mini hole the black of investigation Carlo analytical/Monte discussion. critical a The with qualitative ends interpretation. somewhat talk our the support 5 that Section arguments of events, in Centauro as present the well I of as characteristics a TeV-gravity, basic After in the MBHs main the [1]. of of and scenario features scales relevant above the of the review quick of context the inlcntn,wt udmna gravitational fundamental with constant, tional where n a opt h rvttoa poten- gravitational the compute may One . . 2 lr iheeg olsosi h topee- atmosphere the in collisions energy high ultra n ntesmls aeof case simplest the In are dimensions extra non-compact though Even quantitative more a towards steps first The iibakhls-peitdi e-cl gravity TeV-scale in predicted - holes black mini f + Φ x G R θ , (4) 1 2 i h etna prxmto)of approximation) Newtonian the (in ) ∂ ∂θ ≡ 2 Φ M fCrete, of 2 1 /M = tteoii,b ovn h cor- the solving by origin, the at ∗ − 3 G ste4dmninlgravita- dimensional 4 the is (4) Mδ n ( T x n ) ,oei deal- is one 1, = iheulradii equal with , 2 δ πR ( θ ) (1) 2

scale equal to M∗. Its solution is long as their impact parameter is smaller than the Schwarzschild radius corresponding to their 2πMG(4) 1 − e−2r/R Φ(x,θ)= (2) center of mass energy. A with mass Rr 1 − 2e−r/R cos θ + e−2r/R of the order of the ones discussed here can be where r = |x| and θ is the position in the com- produced in the collision of an ultra high energy pact transverse direction. Near the mass M, primary with atmospheric partons. The energy i.e. for r ≪ R and θ ≪ π it gives Φ ∼ of the primary should exceed 1000 TeV and the (4) 2 2 2 4πMG /(r + R θ ), the correct expression for cross section of the process is conjectured to be a 4 dimensional gravitational potential. At large 2 −37 2 weak σ ∼ πRS ∼ 10 cm ∼ σνN , comparable to distances (r ≫ R) from the mass M, on the the neutrino-nucleon weak interaction cross sec- (4) other hand, one obtains Φ ∼ 2πMG /Rr, the tion at these energies. correct 3 dimensional Newtonian potential with Once produced, black holes are believed to (4) 3 Newton’s constant GN =2πG /R =2π/RM∗ . evaporate. Even though noone has worked out These formulas generalize trivially to n toroidal the details of the evaporation process for such transverse dimensions, in which case Newton’s light black holes, we shall assume the semiclas- constant is given by sical formulas of the standard treatment. So, n+2 n+3 n+3 1 1 within τBH ∼ M∗ (4πRS) /(n + 1) ∼ GN ∼ 2 n (3) −27 M∗ (RM∗) 10 seconds the black hole decays ”democrat- ically” into all kinds of quarks, leptons, gauge Assume that the fundamental scale M of grav- ∗ bosons, gravitons, higgses. It is a fireball of tem- ity is of the order of the weak interaction scale perature T = (n+1)/4πR ∼ 1TeV. The num- M ∼ O(1TeV). Then, (3) leads to the correct BH S ∗ ber of initial emitted by the black hole value G ≃ 10−38GeV−2 for a value of R given N is determined by its entropy N ∼ S = by R ≃ 1/(M (M 2G )1/n). The value n = 1 is initial BH n ∗ ∗ N πM R /2. For the case of black holes with excluded, since it leads to R of the order of the BH S mass of order 1-2 TeV of interest here and n = 4, size of the solar system. For n = 2 one obtains this number is of the order of O(10). R2 ∼ 2mm. Accelerator and astrophysical con- straints lead to a prefered value of n ≥ 4 with a corresponding value for R, much larger than the 4. The Centauro-like events (CLEs) usual value 10−33cm. A normal high energy cosmic ray event is 3. Mini black holes created by the collision of a primary particle with a particle in the atmosphere. Typically a couple As described, the world contains black holes, of leading partons emerge from the interaction generalizations to D = 4+ n spacetime di- region with high transverse momenta, leaving be- mensions of the well known Schwarzschild met- hind a number of soft fragments, mainly pions ric. They are characterized by their mass with relative abundance 1:1:1. The neutral pions MBH , and can exist as long as MBH is subsequently decay to photons and the shower at least a few times the fundamental grav- ends up consisting of low pT < 1 GeV/c parti- ity scale. Lighter black holes cannot exist. cles with Nhadron/Nem ∼ 1 or even less, if one Thus, it is reasonable to expect that for M∗ takes into account the extra photons that will be a fraction of a TeV, the black hole produced as the shower develops even further. are MBH ≥ 2TeV. If, in addition, MBH ≪ In contrast to the above picture, several events 2 −(n+1)/n M∗(GN M∗ ) , the black hole is essentially have been observed since 1972 with the following 4+n-dimensional, since its Schwarzschild radius main characteristics [4]: (a) They were claimed −1 (1/n+1) RS ∼ M∗(M∗/MBH ) ∼ M∗ is much to have taken place at distances smaller that 500 smaller than the radius of the large extra di- m above the detectors at Pamir and Chacaltaya. mensions. Black holes are expected to be pro- It should be pointed out, however, that the alti- duced in the collision of any two particles, as tude was measured directly only for one of these 3 events, and even that has been questioned re- ing the evaporation of the MBH. cently [5]. (b) They are hadron rich, with typ- • Production rate. Assuming that the black ically Nhadron/Nem ≫ 1, (c) have fragments with holes are produced by primary neutrinos, one may high pT ≫ 1 GeV/c, (d) have in many cases obtain a rough estimate of the number of CLEs a heavy central core with tiny angular opening based on current figures for the neutrino flux Φν . (halo) and finally, (e) have all been observed with Since we are interested in neutrino energies of or- energies above a threshold around 500 TeV in the der 106 − 107GeV in the lab frame, one may use lab frame. the estimates for the gamma-ray burst muon neu- It should be pointed out that there are severe trino flux given in [6,7]. Their analysis leads to uncertainties in the observational data. One of ∼20 neutrino-induced muon events in a km3 wa- the speakers in this meeting presented a reanaly- ter or ice per year. Since the cross section of sis of the Centauro I and raised serious doubts MBH production by neutrinos is of the same or- about the altitudes reported in general for all der of magnitude as that of muon production, we these events [5]. Others express doubts about the would expect approximately the above number existence of these events altogether, worrying, for of black holes for each kind of neutrino. Multi- instance, about the fact that no such events have plying by 10-20 (the number of initial jets) and been observed yet in Kanbala and Fuji. We shall taking into account the lower density of the at- not take part in this debate at this point. Instead, mosphere, where the centauros are produced, one we shall assume that the events are real and try to ends up with the estimate of about 10-100 events interpret them as due to evaporating MBHs, pro- per km2 per year, which is one to two orders of duced by ultra high energy E1 > 1000TeV cosmic magnitude smaller than the claimed intensity of ray primaries. the Centauros [4]. However, it seems to us that there is considerable uncertainty in the neutrino flux, which could take care of this discrepancy [8]. 5. CLEs as evaporating mini black holes • Decay products, pT ,Nhadron/Nem. The black The processes of black hole creation, of its sub- hole, depending on its mass, decays initially to sequent evaporation and, finally of the shower 10-20 fundamental particles of all kinds and with formation in the atmosphere involve all the com- equal probability. Their energies are ∼100 GeV plications of several different fields, such as cos- each in the black hole frame. The MBH emits mic ray physics, quantum gravity/string the- almost as a black body of temperature ∼1 TeV, ory, quantum field theory in curved spacetimes, all types of matter and force quanta of the Stan- -gluon plasma physics in QCD, low en- dard Model. The simplest possibility is that ergy non-perturbative QCD, cosmic shower at- these initial partons, with pT s also around 50- mospheric physics, of theoretical and experimen- 100 GeV will form hadrons, mostly mesons, of tal high energy physics and astrophysics. Given all kinds. The charmed or heavier mesons will the incomplete knowledge in all these, a lot has to decay to lighter ones, the neutral pions will de- be done before one can safely confront the obser- cay to photons almost immediately, but the kaons vational data. Nevertheless, we shall make a few will survive a distance of a few hundred meters. simplifying assumptions, implicit in the discus- Since the Ks are counted as hadrons, the ratio sion below, and present the arguments in favour of Nhadron/Nem will be enhanced, if the observa- of the scenario proposed in [1,2,3]. tion takes place at less than a few hundred meters • Energy threshold. The energy threshold of from the initial interaction. The total multiplic- the observed Centauro events, corresponds in the ities of the final showers in this case are a few center of mass frame to a mass roughly a few decades [2]. times M∗ ∼ 1TeV. This coincides with the lower An alternative possibility is that a hot DCC bound on the MBH masses, mentioned above. forms before hadronization of the produced par- The agreement may be even better, if one takes tons. This is known to lead to larger numbers of into account the energy losses into the bulk dur- heavier mesons, which makes it more probable to 4 obtain a large ratio of the hadronic to the elec- Several alternative proposals have been put for- tromagnetic components. However, it is unclear ward to explain these mysterious events [4]. An if the system passes through such a DCC state. In effort was made in [1] and [9] to compare the vari- any case, it seems that the probability to obtain ous scenaria. It should be clear however, that the a superclean Centauro event is rather small. study, for instance, of the production and evap- On the other hand, given the typically large oration processes of a MBH or of Strange Quark values of pT , the present scenario seems to be the Matter fireballs, of the potential formation of a most natural one to explain this feature of the high temperature quark-gluon plasma phase or of Centauros [9]. a DCC, of the hadronization of the 100 GeV par- • Deep penetration. The deep penetration fol- tons, all rely on unknown aspects of the physics lows from the assumption of neutrinos, or some involved. Much more work is necessary and all other weakly interacting particle (WIMP?), as possible realizations of a given scenario have to the primary source of these events. According to be investigated, before one can safely fit the ob- the present picture any black holes produced at servational data. The fundamental importance of higher altitudes from the detectors, will give sig- the issues involved deserves every effort. nals similar to standard events [2]. This kind of signature with large hadronic to electromagnetic REFERENCES ratio, could only be obtained from the decay of black holes near the detector. 1. ”Can Centauros or Chirons be the first ob- • Halo. At a more speculative level, it has servations of evaporating mini black holes?”, been claimed recently [10] that after evaporation A.Mironov, A.Morozov and T.N.Tomaras; a MBH is expected to leave behind a highly ex- hep-ph/0311318. cited string state (a string ball), which in princi- 2. A.Cafarella, C.Corian`o and T.N.Tomaras, “Cosmic Ray Signals from Mini Black Holes ple will decay to light (compared to M∗) particles. Their life-time depends strongly on the excitation in models with Extra Dimensions: An Ana- −1 lytical/Monte Carlo study”, hep-ph/0410358. level and can be considerably larger than M∗ . These objects may be serious candidates in the 3. A.Cafarella, C.Corian`o and T.N.Tomaras, context of the present scenario for the projectiles “Searching for Extra Dimensions in High En- responsible for the halos observed mostly in Chi- ergy Cosmic Rays”, hep-ph/0410190. Talk rons. presented in this meeting by Claudio Corian`o. 4. E.Gladysz-Dziadu´s, Phys. Part. Nucl. 34 (2003) 565; hep-ph/0111163, and references 6. Discussion therein. 5. V.Kopenkin, A.Ohsawa, E.Shibuya and On the basis of the above qualitative presen- M.Tamada, Proc. 28th International Cos- tation and of the results of the first numeri- mic Ray Conf., 2003; p.1583. V.Kopenkin, cal steps taken in [2,3], it seems that the above Y.Fujimoto and T.Sinzi, Phys. Rev. D68 scenario may account quite succesfully (a) for (2003) 052007. the energy threshold of all these events of order 6. E.Waxman and J.N.Bahcall, Phys. Rev. D59 O(TeV) in the center of mass frame, (b) the to- (1999) 023002; hep-ph/9902383. tal multiplicities of a few dozen particles, depen- 7. E.Waxman, Nucl. Phys. Proc. Suppl. 118 dent on the number of extra dimensions [2], (c) (2003) 353. the large values of p , and (d) the number and T 8. O.E.Kalashev, V.A.Kuzmin, D.V.Semikoz the rough heights of first interaction of the Cen- and G.Sigl, Phys. Rev. D66 (2002) 063004. tauro/Chiron events. It seems less succesfull in 9. E.Gladysz-Dziadu´s, hep-ph/0405115. giving the right N /N (especially for the hadron em 10. D.Chialva, R.Iengo and J.Russo, superclean events), even though due to large sta- hep-th/0310283. tistical fluctuations, the numerical simulations [2] have not been conclusive yet. arXiv:hep-ph/0411081v1 5 Nov 2004 \addressmark[MCSD] iha pinlagmn othe to argument optional an with ‡ \addressmark † inlagmn othe to argument tional e.g. leirisrcin o h rprto fa2clm for 2-column L a in of paper preparation camera-ready the for instructions Elsevier drs) eerhgat pnoigaec,ec These etc. normal the agency, with your obtained sponsoring from are different grant, (if research address address), present your cate ∗ .d Groot de P. ic od icetr aphr 3 1,Uie Kingdom United T19, P3L Hampshire Winchester, Road, Finch 2 Netherlands The b Amsterdam, AC 1000 103, Box P.O. a dp uigtepeaaino orppr hsi h outpu the is This paper. your of preparation the during adopt edn otetpo h etpg;(i L (ii) page; next the the of transfers top but page, the a to of heading bottom the at rectly ae eghaatfo h olwn w excep- two L following (i) the tions from apart length lated n 1c nscn n olwn ae.The pages. following and pages second first cm on on L 16 cm cm 19.5 of 21 of width and length total maximum wide margin, a cm middle and 7.5 cm column 1 each with pages: these on shown FORMAT 1. l)ln pcn.Hwvr hntpn com- typing L when text However, mathematical plicated spacing. line gle) Spacing [1,2]. 1.1. references: or some text of are section Here a paragraph. complete a to order the of in length area the text exceeds ever) hardly (well, never nrae h pc ewe etlnsi order overlapping in fonts lines superscript and text sub- prevent between to space the increases A oruea drsmr ae n ae h address the label on, later the addressmark use an address reuse same To the with authors following For otoe hudapa ntefis aeol oindi- to only page first the on appear should Footnotes ahmtc n optrSineScin leirSineB.V., Science Elsevier Section, Science Computer and Mathematics cnmc eatet nvriyo Winchester, of University Department, Economics T hs ae rvd o iha xml ftelyu n styl and layout the of example an with you provide pages These etsol epoue ihntedimensions the within produced be should Text enral eomn h s f10(sin- 1.0 of use the recommend normally We E ouetcasue h aiu stipu- maximum the uses class document X \address[MCSD] A T a E command. ∗ osntbgnanwscindi- section new a begin not does X .d Maas de R. , . n eettelbla h op- the as label the repeat and , \thanks \addressmark command. a † .Y Wang X.-Y. , A T E \address automatically X A T omn,e.g. command, E X command, b A n .Sheffield A. and T E X 1 t,tecnrs ftx etrn utb uniform, be must lettering text qual- and of reproduction contrast the heads optimal ity, achieve text To between and bottoms. or left-hand margins between considerable right-hand has either or variation streaks, lighting dark and light tains which head, the number. before page page the the of contains top the docu- at cm the 3 by class these determined to ment are changes which make Letter) not dimensions, US do (or to: adhered A4 strictly white use size. plain make paper quality Please good acceptable. of not is photocopy a (“letter- printhead pin 24 or 18 quality”). used an be possesses only it should printer if matrix dot A printer. PRINTOUT 2. and Light Bookman Other Roman, Palatino. Schoolbook, Times Century point Roman. 10 New Modern are Computer fonts recommended point 10 a Fonts 1.2. il- matter printed your making legible. and another one rnessmtmspouetx hc con- which text produce sometimes Printers original; an be should submitted printout The inkjet an or laser a is printer suitable most The using produced been have instructions These rmteL the from t a ‡ o 0%rpouto hc ews o to you wish we which reproduction 100% for e h ouetcaslae tleast at leaves class document The . h iesossonhr hudbe should here shown dimensions The A T E ouetcasyurequested. you class document X mat 2 sharp and dark over the whole page and through- out the article. If corrections are made to the text, print com- pletely new replacement pages. The contrast on ] ? these pages should be consistent with the rest [ 21 45 . . 0 0 0.03 4.859 3.979 5.76 of the paper as should text dimensions and font 18 sizes. v

3. TABLES AND ILLUSTRATIONS

Tables should be made with LATEX; illustra- − − − − − − − tions should be originals or sharp prints. They − should be arranged throughout the text and preferably be included on the same page as they are first discussed. They should have a self- 10 44 . . 0 contained caption and be positioned in flush- 0 the pion field. − left alignment with the text margin within the − column. If they do not fit into one column they may be placed across both columns (us- without 12 ing \begin{table*} or \begin{figure*} so that 45 . . 0 they appear at the top of a page). 0 − − 3.1. Tables Tables should be presented in the form shown 15 in Table 1. Their layout should be consistent 45 . . 0 throughout. 0 − − Horizontal lines should be placed above and be- low table headings, above the subheadings and at the end of the table above any notes. Vertical 18 45 . lines should be avoided. . 0 0 − If a table is too long to fit onto one page, − the table number and headings should be re- peated above the continuation of the table. For this you have to reset the table counter with 19 45 . . 0 \addtocounter{table}{-1}. Alternatively, the 0 − table can be turned by 90◦ (‘landscape mode’) − and spread over two consecutive pages (first an even-numbered, then an odd-numbered one) Table 2: The next-to-leading order (NLO) results 19 created by means of \begin{table}[h] without 45 . . 0 0 a caption. To do this, you prepare the table as 0.867 0.864 0.855 0.845 0.834 0.823 0.8574 0.847 − − a separate LATEX document and attach the tables to the empty pages with a few spots of suitable glue. ) 0.259 0.268 0.287 0.302 0.312 0.319 0.286 0.270 2 (fm) 4.887 4.881 4.864 4.846 4.827 4.810 (fm) 3.995 3.989 3.973 3.955 3.936 3.918 (%) (%) 0.03 0.03 0.03 0.03 0.03 0.03

3.2. Useful table packages (%) (%) 2.32 2.83 4.34 6.14 8.09 9.90 (fm (fm) 1.973 1.972 1.974 1.978 1.983 1.987 1.966(7) 1.967 GT A M1

Modern LT X comes with several packages for d D E d C2:N C2:C VP 1B 1B 1B d δ δ δ M µ M Λ(MeV)r 140 150 175 200 225 250 Exp. Q tables that provide additional functionality. Be- P low we mention a few. See the documentation

of the individual packages for more details. The The experimental values are given in ref. [4]. 3

Table 1 The next-to-leading order (NLO) results without the pion field. Λ(MeV) 140 150 175 200 rd (fm) 1.973 1.972 1.974 1.978 2 Qd (fm ) 0.259 0.268 0.287 0.302

PD (%) 2.32 2.83 4.34 6.14

µd 0.867 0.864 0.855 0.845

MM1 (fm) 3.995 3.989 3.973 3.955

MGT (fm) 4.887 4.881 4.864 4.846 VP δ1B (%) −0.45 −0.45 −0.45 −0.45 C2:C δ1B (%) 0.03 0.03 0.03 0.03 C2:N δ1B (%) −0.19 −0.19 −0.18 −0.15 The experimental values are given in ref. [4]. packages can be found in LATEX’s tools directory. figure at the top of a page, the top of the figure should align with the bottom of the first text line array Various extensions to LATEX’s array and of the other column. tabular environments. Do not use too light or too dark shading in your figures; too dark a shading may become too dense longtable Automatically break tables over sev- while a very light shading made of tiny points may eral pages. Put the table in the longtable fade away during reproduction. environment instead of the table environ- All notations and lettering should be no less ment. than 2 mm high. The use of heavy black, bold dcolumn Define your own type of column. lettering should be avoided as this will look un- Among others, this is one way to obtain pleasantly dark when printed. alignment on the decimal point. tabularx Smart column width calculation 3.4. PostScript figures within a specified table width. Instead of providing separate drawings or rotating Print a page with a wide table or prints of the figures you may also use Post- figure in landscape orientation using the Script files which are included into your LATEX sidewaystable or sidewaysfigure envi- file and printed together with the text. Use ronments, and many other rotating tricks. one of the packages from LATEX’s graphics di- Use the package with the figuresright op- rectory: graphics, graphicx or epsfig, with tion to make all tables and figures rotate the \usepackage command, and then use the in clockwise. Use the starred form of the appropriate commands (\includegraphics or sideways environments to obtain full-width \epsfig) to include your PostScript file. tables or figures in a two-column article. The simplest command is: \includegraphics{file}, which inserts the 3.3. Line drawings PostScript file file at its own size. The starred Line drawings may consist of laser-printed version of this command: graphics or professionally drawn figures attached \includegraphics*{file}, does the same, but to the manuscript page. All figures should be clips the figure to its bounding box. clearly displayed by leaving at least one line of With the graphicx package one may spec- spacing above and below them. When placing a ify a series of options as a key–value list, e.g.: 4

\includegraphics[width=15pc]{file} \includegraphics[height=5pc]{file} \includegraphics[scale=0.6]{file} \includegraphics[angle=90,width=20pc]{file} See the file grfguide, section “Including Graphics Files”, of the graphics distribution for all options and a detailed description. The epsfig package mimicks the commands fa- miliar from the package with the same name in LATEX2.09. A PostScript file file is included with the command \psfig{file=file}. Grey-scale and colour photographs cannot be included in this way, since reproduction from the printed CRC article would give insufficient typo- Figure 2. Remember to keep details clear and graphical quality. See the following subsections. large enough.

author(s) of the paper. As costs involved are per page, care should be taken in the selection of size and shape so that two or more illustrations may be fitted together on one page. Please contact the Author Support Department at Elsevier (E- mail: [email protected]) for a price quotation and layout instructions before produc- ing your paper in its final form.

4. EQUATIONS Equations should be flush-left with the text margin; LATEX ensures that the equation is pre- ceded and followed by one line of white space. Figure 1. Good sharp prints should be used and LATEX provides the document class option fleqn not (distorted) photocopies. to get the flush-left effect.

(0) Hαβ(ω)= Eα (ω)δαβ + hα|Wπ|βi (1)

3.5. Black and white photographs You need not put in equation numbers, since Photographs must always be sharp originals this is taken care of automatically. The equation (not screened versions) and rich in contrast. numbers are always consecutive and are printed They will undergo the same reduction as the text in parentheses flush with the right-hand margin and should be pasted on your page in the same of the text and level with the last line of the equa- way as line drawings. tion. For multi-line equations, use the eqnarray environment. 3.6. Colour photographs For complex mathematics, use the AMSmath Sharp originals (not transparencies or slides) package. This package sets the math indenta- should be submitted close to the size expected tion to a positive value. To keep the equations in publication. Charges for the processing and flush left, either load the espcrc package af- printing of colour will be passed on to the ter the AMSmath package or set the command 5

\mathindent=0pt in the preamble of your arti- cle.

REFERENCES 1. S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222. 2. O.V. Mazurin and E.A. Porai-Koshits (eds.), Phase Separation in Glass, North-Holland, Amsterdam, 1984. 3. Y. Dimitriev and E. Kashchieva, J. Mater. Sci. 10 (1975) 1419. 4. D.L. Eaton, Porous Glass Support Material, US Patent No. 3 904 422 (1975).

References should be collected at the end of your paper. Do not begin them on a new page unless this is absolutely necessary. They should be prepared according to the sequential numeric system making sure that all material mentioned is generally available to the reader. Use \cite to refer to the entries in the bibliography so that your accumulated list corresponds to the citations made in the text body. Above we have listed some references according to the sequential numeric system [1,2,3,4].