CONFERENCE PROCEEDINGS
International Student Conference
“Science and Progress”
German-Russian
Interdisciplinary
Science Center
St. Petersburg – Peterhof
September, 30 - October, 4
2013
Organizing committee
Prof. Dr. A.M. Shikin, G-RISC Coordinator, SPSU
E. Serova, G-RISC office, SPSU
T. Zalialiutdinov, G-RISC office, SPSU
M. Rusinova, G-RISC office, SPSU
Program Committee
Prof. Dr. A.M. Shikin, G-RISC Coordinator, SPSU
Prof. Dr. E. Ruehl
G-RISC Coordinator, Freie University Berlin
Dr. A.A. Manshina, Faculty of Chemistry, SPSU (Section A - Chemistry)
Prof. Dr. V.N. Troyan, Faculty of Physics, SPSU (Section B – Geo- and
Astrophysics)
Prof. Dr. N.R. Ikhsanov, Faculty of Mathematics and Mechanics, SPSU
(Section B – Geo- and Astrophysics)
Prof. Dr. V. Reitmann, Faculty of Mathematics and Mechanics, SPSU
(Section C – Mathematics and Mechanics)
Prof. Dr. A.P. Baraban, Faculty of Physics, SPSU (Section D – Solid State
Physics)
Dr. A.S. Chirtsov, Faculty of Physics, SPSU (Section E – Applied
Physics)
Prof. Dr. N.A. Timofeev, Faculty of Physics, SPSU (Section F – Optics and
Spectroscopy)
Prof. Dr. Yu.M. Pismak, Faculty of Physics, SPSU (Section G – Theoretical,
Mathematical and Computational Physics)
Prof. Dr. N.V. Tsvetkov, Faculty of Physics, SPSU (Section H - Biophysics)
Dr. M.G. Sheliapina, Faculty of Physics, SPSU (Section I – Resonance
Phenomena in Condensed Matter)
Contacts
Faculty of Physics, Saint-Petersburg State University
Ulyanovskaya ul. 1,
Peterhof, St. Petersburg, Russia
198504
Tel. +7 (812) 428-46-56,
Fax. +7 (812) 428-46-55
E-mail: g-risc@phys.spbu.ru
Website: www.g-risc.org
3
A. Chemistry
Development of the method for determining the reaction
products of organic compounds occurring in laser-induced
deposition of copper from solution
Zhigley Elvira, Safonov Sergey
eh157@yandex.ru
Scientific supervisor: Dr. Kochemirovsky V.A. Department of Laser
Chemistry and Laser Materials Science, Faculty of Chemistry,
Saint-Petersburg State University
Laser-induced deposition of copper from solution (LCLD) is the method based
on the reaction of metal recovery flowing in local volume of solution in the laser
beam focus, resulting in precipitation of the metal on the surface of the dielectric
substrate. High intensity of focused radiation, especially using impulse lasers, creates
local non equilibrium states with high temperature and concentration gradients.
The main aim of the research was to develop the method for determining
products of non equilibrium reactions and to study their influence on copper precipitate topology.
LCLD reaction was provided on surface of dielectric Glass-ceramics substrate.
Composition of solution С(CuCl2*2H2O) = 0.01 M; C( C4H2O4KNa*4H2O) =
0.03M; С(NaOH) = 0.05 M. Deposition was during 8, 10,12 hours. Solutions
after irradiation were extracted with ch2cl2 and this extract was investigated by
the method of gase chromatography mass spectrometry. Chromatography-mass
spectra which contain large amounts of difficult identifiable peaks were obtained.
It indicates the presence of a large number of the inorganic components as the
products of the reactions. The difficulties of identification associate with a large
number of overlapping observed peaks.
As a result, we can conclude that the selected method is not suitable for determining the reaction products laser-induced deposition of copper from solution.
Further researches may be carried out by reversed-phase chromatography.
Fig. 1. Chromatogram of solution solutions after irradiation 8 h.
References
1. Tver’yanovich Y.S., Kuzmin A.G, Menchikov L.G., Kochemirovsky V.A.,
Safonov S.V., Tumkin I.I., Povolotsky A.V., Manshina A.A. // Mendeleev
Communications, v. 21, № 1, pр. 34-35 (2011).
6
D. Solid State Physics
Characterization of pristine and fluorinated
nanodiamonds by X-ray absorption spectroscopy
Zagrebina Elena
zagrebina.phys@mail.ru
Scientific supervisor: Prof. Dr. Vinogradov A.S., Department of
Solid State Electronics, Faculty of Physics, Saint-Petersburg State
University
Introduction
At present time carbon nanostructures have been actively investigated. Among
them there are nanodiamonds (NDs) – crystalline diamond nanoparticles with a size
of 4÷150 nm. They attract the particular attention of scientists due to their unique
properties, such as high thermal conductivity, the highest hardness, large specific
surface area, high refraction and special mechanical antifrictional characteristics.
The NDs powders are used can be useful for different applications in a wide variety
of fields: in medicine as drug delivery, in mechanics as additives and lubricants, in
technology of creating and polishing of hard materials, in electronics and biology,
etc. [1, 2]. However, chemical inertness of NDs makes it difficult to with them.
Therefore various functionalization methods of NDs are used to activate their
surface. One of the most effective ways of such activation is surface fluorination.
However the mechanism of this process and its impact on surface chemistry are
still not fully understood.
The main purpose of this work was to clarify the possibility of using near-edge
x-ray absorption fine structure (NEXAFS) spectra at the C 1s and F 1s absorption
edges for characterization of pristine and fluorinated nanodiamonds (NDs and
F-NDs) and to obtain information on their atomic and electronic structures as
well as on the character of chemical bonding between carbon and fluorine atoms
for F-NDs. The measured NEXAFS spectra were analyzed by comparing with
the spectra those of reference systems – carbon nanodiscs CNDs and graphite
monofluoride (CF)n [3].
Experimental
The diamond materials studied in this work were the following ones: pristine
nanodiamonds (Raw-ND, ND4, ND30, and ND50) synthesized by different ways
and their fluorinated derivatives (Raw-ND-F, ND4-F, ND30-F, and ND50-F).
All the samples for investigation were given by Prof. Dr. M. Dubois (Clermont
Université, Université Blaise Pascal, France). Synthesis methods, treatment types
and average diameter of NDs particles for every substance are presented in Table 1.
The pristine NDs particles were fluorinated using flux of pure F2 (1 atm.) at 500°C
for 12 h [1]. All samples for measurements were prepared ex situ in air by rubbing
the powders of NDs into previously cleaned and scratched surface of copper plate
5x5 mm2 in size in order to ensure the substrate uniform surface coating without
noticeable gaps.
8
Table 1. Characteristics of the pristine NDs studied: names, sizes, sythesis and
treatment methods.
NDs
Average
Synthesis method and treatment
diameter
Raw-ND
different
ND4
4 nm
ND30
30 nm
ND50
50 nm
Detonation before disintegration
Detonation, acid treatments and disintegration with
large excess of ZrO2
Size-separating the powdered diamond from the
bulk synthesized by a static high-pressure high-temperature (HPHT) method, before hydrogen-plasma
treatment, the HPHT NDs were treated in a heated
solution of acids
Summarizes the synthesis methods: High Pressure
High Temperature (HPHT) and Shock Compression
method (SC)
All measurements were carried out using monochromatic synchrotron radiation
and the facilities of the at D1011 beamline (BL) at the electron storage ring MAX
II (the MAX IV laboratory, Lund University, Sweden) [4]. This BL is based on the
modified Zeiss SX-700 plane grating type monochromator with 1200 mm-1 groove
density grating which covers a range of photon energies from 30 to 1500 eV with
high energy resolution. The BL D1011 is equipped with two experimental endstations: front (spectroscopic) and back (magnetic), the last can be used for x-ray
absorption study of the various materials, including magnetic ones. This experiment
was carried out using the measuring chamber of the back endstation, the samples
were transferred there from air and fixed on sample holder. The samples of studied
NDs and F-NDs were annealed at temperature of 350 °C for 30 minutes before
measurements to eliminate any possible influence of adsorbed carbon contaminations of the substrate on the C 1s NEXAFS spectra. Previously, it was found that
such annealing of a copper substrate substantially removes carbon contaminations
from its surface. Indeed, the almost complete disappearance of low-intense C 1s
absorption structures, which are observed in the spectrum of the substrate at room
temperature, occurs after the similar annealing. All absorption spectra were measured under ultrahigh vacuum conditions with residual gas pressure in the experimental chamber ~10-9 mbar. The samples were located at an angle of ~ 45° with
respect to the incident beam of monochromatic radiation and the size of the focal
spot on the sample was around 2×1 mm2. No sample charging or decomposition
effects upon its irradiation by an intense beam of soft x rays were detected during
measurements. The spectra were recorded several times from different points of
the sample and their structure had usually a good reproducibility.
The NEXAFS spectra at the C 1s and F 1s thresholds of the NDs and F-NDs
were measured by recording the total and partial electron yield (TEY and PEY) of
X-ray photoemission as a function of the incident photon energy. It is well known
9
that the electron field near the x-ray absorption edge is proportional to the absorption cross section [5,6]. The TEY spectra were acquired by measuring the drain
current from the sample, while the PEY spectra were recorded by the multiplechannel plate detector with the retarding voltage Vret = -150 V. These techniques
for measuring X-ray absorption are characterized by a probing depth of >10 nm
and ~ 0.5 – 0.8 nm, respectively. All high-velocity Auger electrons, slow secondary electrons and photoelectrons are detected when measuring TEY spectra [6].
In this case the large probing depth is due to a large escape depth of slow secondary electrons which have a large mean free path. Using of the retarding negative
potential for PEY spectra registration, the slowest secondary and photoelectrons
are cut off, thus providing the probing depth <1 nm. That’s why absorption spectra
measured with PEY technique are more surface sensitive [6].
All absorption spectra were recorded with the exit slit size 16 μ, that provided
the photon-energy resolution set to 120 meV for C 1s edge (photon energy
hν ~ 285 eV) and 350 meV for F 1s edge (photon energy hν ~ 685 eV). The
X-ray absorption spectra were normalized to the incident photon flux, which was
monitored by recording the TEY from the clean surface of a gold mesh mounted
in the beamline (in the front of the sample). The photon energy hν over the range
of F and C 1s absorption spectra fine structure was calibrated against the energy
positions of the first narrow peak in the F 1s X-ray absorption spectrum of solid
K2TiF6 (F 1s→ t2g, 683.9 eV [7]) and the C 1s X-ray absorption spectrum of solid
C60 (C 1s → LUMO, 284.5 eV [8]).
Results and discussion
C 1s absorption spectra for pristine NDs annealed at 350°C for about 30 min
are presented in Fig. 1 together with the corresponding one of graphitized carbon
nanodiscs from Reference [3]. Spectra were simultaneously recorded in TEY and
PEY modes (red and blue
curves respectively).
From Fig. 1 it is clear
that C 1s spectra of all NDs
have similar spectral shape
with the main absorption
bands B* - F* that are characteristic of the crystalline
diamond spectrum [9] and
some additional low-energy
structures A and A1. It should
be noted that in the spectra
of all the NDs except ND30
there is no sharp peak B at an
Fig. 1. C 1s absorption spectra for pristine NDs and energy of ~ 289 eV, which is
carbon nanodisks (CNDs) [3]. The spectra are nor- observed in the spectrum of
malized to the same level of continuous 1s absorp- crystalline diamond and is
tion at the photon energy of 320 eV.
attributed to the σ exciton
10
[9, 10]. Interestingly, this exciton is usually not observed in the spectra of nanodiamonds [11]. It is possible that the presence of the σ exciton B in the spectrum
ND30 is due to its better crystallinity compared to other nanodiamonds, which in
turn is associated with the features of synthesis method (HPHT) used for ND30
preparation.
Only very small changes are observed in the spectral shape of bands B* - F*
along the ND50-ND30-ND4 series, while the relative intensities of additional bands
A and A1 are greatly reduced with decreasing size of the NDs particles. Furthermore,
the structures A and A1 are more pronounced in the PEY spectra in comparison with
the TEY ones. Taking into account surface sensitivity of the PEY mode (probing
depth < 1 nm), this finding indicates that bands A and A1 may correspond to the C
1s electron transitions to the unoccupied states of carbon atoms on the NDs surface.
On the other hand, the surface contribution to the total signal is expected to grow
with decreasing the diamond particle size, whereas the structures A and A1 on the
contrary decrease in intensity. Therefore, they cannot be related to the surface carbon
atoms of NDs and should be attributed to the carbon atoms of some foreign carbon
clusters on NDs surface. Probably the number of such clusters depends on the NDs
synthesis method and the NDs particles size, thus causing a variation of peaks A
and A1 in intensity for different samples. From Fig. 1 it is evident that the energy
position of the structure A (285.0 eV) is very close to the position of resonance A
in the C 1s spectrum of graphitized carbon nanodiscs CNDs (~ 285.1 eV), which
is characteristic of sp2 coordinated carbon atoms. A small low-energy shift of A in
going from CNDs to NDs may be attributed to amorphization of carbon clusters
[12]. Thus, the structure A corresponds to the C 1s electron transitions to unoccupied C 2p electron states in some amorphous sp2-like carbon clusters on the NDs
surface. In its turn, structure A1 (286.9 eV) can be associated with oxidized carbon
atoms of these clusters [13, 14]. Thus, as-prepared NDs particles have on their
surface some amorphous
carbon clusters, which are
partially oxidized.
Now let us examine influence of fluorination process on the spectra of NDs
by the example one of them.
In Fig. 2 the C 1s absorption
spectra for samples of pristine ND50 and fluorinated
ND50-F are compared. It is
well seen that the absorption
bands B*- F* characteristic
of diamond scarcely do not
change in going from the
pristine ND to the fluorinated Fig. 2. C 1s absorption spectra for pristine and fluone, while the additional orinated ND50.
11
structures A and A1 are significantly reduced in their relative intensity. In addition,
the new structure B1* at photon energy of 289.4 eV appears after fluorination in
the range of absorption band B*. Clearly the fluorination of ND50 is responsible
for changes observed in the spectra under comparison. As mentioned above, the
structures A and A1 reflect absorption transitions of the C 1s electrons in the atoms
of surface carbon clusters. A decrease in the relative intensity of the structures A
and A1 in the spectrum ND50-F indicates that the number of surface carbon clusters
decreases during the ND50 fluorination. This is possible only if the volatile compounds of carbon, oxygen and fluorine atoms are formed on the surface of diamond
nanoparticles during their
fluorination. In turn, the
emergence of structure B is
apparently caused by covalent attachment of fluorine
atoms to carbon atoms of the
diamond nanoparticles.
Similar changes in the C
1s spectra are also observed
in the spectra of other fluorinated NDs (ND-Fs) presented in Fig. 3. It can be seen
that intensity suppression
of the structures A and A1 in
Fig. 3. C 1s absorption spectra for fluorinated NDs absorption spectra of other
and (CF)n [3].
NDs is even stronger than
in the ND50 spectrum: the
structures A and A1 practically disappear and are
hardly visible in the spectra.
Apparently it means that the
fluorination almost completely cleans the surface
of NDs particles from the
carbon and carbon-oxygen
clusters. It seems plausible
that the volatile compounds
of carbon atoms with oxygen and fluorine atoms are
formed and then they leave
the surface of the diamond
Fig. 4. F 1s absorption spectra for fluorinated NDs nanoparticles.
In Fig. 3 C 1s absorpand (CF)n [3]. The spectra are normalized to the
same level of continuous 1s absorption at the photon tion spectrum of graphite
fluoride (CF)n from [3] is
energy of 725 eV.
12
also presented for a comparison with the spectra of ND-Fs. As seen from Figure,
this spectrum is very similar in spectral profile to the NDs spectra. However, it
is more structured over the range of absorption structures B*-F* in comparison
with the NDs spectra and shows an intense peak B1*, which is of low-intensity in
the spectra of NDs. From the ND30-F spectrum it is seen that this peak is in the
region of the C 1s exciton of the crystalline diamond. The fundamental reason of
the similarity of the spectra under comparison is the close tetrahedral coordination
of carbon atoms in ND-Fs and (CF)n, that results from a covalent σ(C-F) bonding between the fluorine and carbon atoms [3]. Each of carbon atoms has the C3F
coordination in the graphite fluoride (CF)n, while most of carbon atoms in ND-Fs
are surrounded by carbon tetrahedron (C4) and only the surface carbon atoms have
the C3F coordination. This is the reason of a more contrast of the absorption bands
B*-F* in the spectrum of (CF)n as compared to the ND-Fs spectra.
The σ bonding effects in (CF)n are represented by an intensity of the absorption
peak B1* [3]. This structure is associated with the C 1s electron transitions to the
unoccupied states in carbon atoms that are bonded with fluorine atoms. An energy
correlation of B1* in the spectra of ND-Fs and (CF)n allows to make a conclusion
about covalent bonding of the fluorine and carbon atoms on the NDs surface during
the functionalization (fluorination) process.
An analogous conclusion can be drawn considering F 1s absorption spectra
for NDs-F with corresponding spectrum of graphite fluoride (CF)n (Fig. 4). All the
compared spectra have a similar spectral shape with a single intensive absorption
band B1*. It is important that the latter has a close energy position in all the spectra
and evidently reflects the C 1s electron transition in the fluorine atoms that are
covalently bonded with surface carbon atoms of ND-Fs. It can be seen directly
comparing the F 1s spectra of ND-Fs and (CF)n.
Conclusions
To sum up, NEXAFS C 1s and F 1s spectra of pristine and fluorinated diamond
nanoparticles of different size 4 nm, 30 nm, and 50 nm were measured with a
high-energy resolution using synchrotron radiation of the beamline D1011 at the
electron storage ring MAX II (Lund University, Sweden). The spectra obtained
were analyzed by comparing with the corresponding ones of reference systems
– carbon nanodiscs CNDs and graphite monofluoride (CF)n. It was found that
diamond nanoparticles have on their surface some amorphous carbon clusters,
which are partially oxidized. According to the measured C 1s spectra of NDs-F, the
surface fluorination removes these clusters and carbon-oxygen groups from NDs
surface forming possibly volatile compounds. A comparison C 1s and F 1s spectra
of F-NDs with corresponding spectra of the fluorinated graphite (CF)n points to
similar character chemical bonding between fluorine and carbon atoms and close
tetrahedral coordination of carbon atoms in these compounds. In the fluorination
process the fluorine atoms are bonded with surface carbon atoms of NDs and form
covalent σ(C-F) bonds that are similar to the ones in (CF)n.
13
This work was supported by the St. Petersburg State University (Grant No.
11.38.638.2013), the Russian Foundation for Basic Research (Grant Nos.12-0200999 and 12-02-31415).
References
1. Dubois M., Guerin K., Batisse N., Petit E., Hamwi A., Komatsu N.,
Kharbache H., Pirotte P., Masin F. // Solid State NMR 40 (2011) 144.
2. Krueger A., Lang D. // Advanced Functional Materials, 22 (2012) 890-906.
3. Ahmad Y., Dubois M., Guérin K., Hamwi A., Fawal Z., Kharitonov A.P.,
Generalov A.V., Klyushin A.Yu., Simonov K.A., Vinogradov N.A., Zhdanov I.A.,
Preobrajenski A.B., Vinogradov A.S. // J. Phys. Chem. C, 117 (2013) 13564.
4. Nyholm R., Svensson S., Nordgren J., Flodström A. // Nucl. Instr. and Meth.
A (1986), 246, 267-271.
5. Lukirskii A.P., Brytov I.A. // Fiz. Tverd. Tela, 6 (1964), 43.
6. Gudat W., Kunz C. // Phys. Rev. Let., 3 (1972) 29, 169-172, 7.
7. Vinogradov A.S., Fedoseenko S.I., Krasnikov S.A., Preobrajenski A.B.,
Sivkov V.N., Vyalikh D.V., Molodtsov S.L., Adamchuk V.K., Laubschat C., Kaindl
G. // Phys. Rev. B, 71 (2005) 045127.
8. Brühwiler P.A., Maxwell A.J., Nilsson A., Whetten R.L., Mårtensson N. //
Chem. Phys. Lett., 193 (1992) 311-316.
9. Morar J.F., Himpsel F.J., Hughes J.L., McFocly F.R. // Phys. Rev. B, 33, 2
(1986) 1346-1349.
10. Morar J.F., Himpsel F.J., Hollinger G., Hughes G., Jordan J.L. // Phys. Rev.
Lett., 54 (1985).
11.Liu X., Klauser F., Memmel N., Bertel E., Pichler T., Knupfer M., Kromka A.,
Steinmuller-Nethl D. // Diamond Relat. Mater., 16 (2007) 1463-1470.
12.Gago R., Vinnichenko M., Juger H.U., Belov A.Yu., Jimenez I., Huang N.,
Sun H., Maitz M.F. // Phys. Rev. B (2005) 72.
13.Sekiguchi T., Ikeura-Sekiguchi H., Baba Yu. // Surf. Sci. 454–456 (2000),
363–368.
14.Kuznetsova A., Popova I., Bronikowski M.J., Huffman C.B., Liu J.,
Smalley R.E., Hwu H.H., Chen J.G. // JACS 123 (2001) 10699–10704.
14
E. Applied Physics
Migration of the radionuclides from nuclear accidents in
the forest ecosystem
Merzlaya Anastasia
stummeworte@mail.ru
Scientific supervisor: Dr. Sergienko V.A., Department of Nuclear
Physics, Faculty of Physics, Saint-Petersburg State University
Introduction
Nuclear catastrophes significantly affect the level of the radioactive contamination. The Chernobyl nuclear power station explosion in 1986 was the worst radiation
catastrophe in history of atomic energy. The result of the recent accident on nuclear
power plant occurred in Fukushima in 2011. The result of this was spreading released radionuclides (more then 10 thousand PBq) from discarded nuclear reactor
materials all over the world. Residual radiation affects the level of pollution of the
environment of St. Petersburg and Leningrad region.
The aim of the work was investigation of the influence of the accidents on the
pollution of the environment and research the radionuclide migration in the environment of the territory of Saint-Petersburg and Leningrad region on the example
of the accident at the Chernobyl NPP and Fukushima NPP.
The method of investigation
Soon after the Chernobyl accident the investigation was carried out to notify the
pollution of Europe by gamma-spectrometric survey from planes using scintillation
detectors. Although this method gives uncertain data of radionuclide activity in the
study area, it allows obtaining an overall picture of the contamination. Thus, according the research in the Leningrad region was found 137Cs from Chernobyl accident
in Kingisepp and Lomonosov districts with an estimated activity 1–5 Ci/km2 [1].
Another method of investigation of the contaminated territories is the collection of samples of some objects of an environment and measuring them in
the laboratory using semiconductor High Pure (HP) Ge-detector. This method
of registration of radioactivity requires a longer time, but gives information that
is much more reliable. Thus, according the earliest researches with soil samples
[2] it was notified that contamination after the accident at the Chernobyl NPP is
larger than it was established earlier and radioactive spots are located throughout
all Leningrad region. However, Leningrad region is not appreciably affected by
the contamination from Fukushima.
So for this investigation territory with not very high radioactivity of the environment to be sure that caesium has Chernobyl or Fukushima origin and not very
low radioactivity to have a possibility to detect impact. For this aims samples from
Pavlovsk territory were chosen to measure.
To investigate the influence of the accidents on the pollution of environment
the forest ecosystem was studied. In such an ecosystem the anthropogenic impact
is rather poor, so that it enables to research the natural processes of radionuclide
16
migration in the environment. Also it permits to inquire into the number consumed
radionuclides with the products of forest by population.
In order to obtain a complete picture of the radionuclides position from accidents
in such an ecosystem, we measured environmental objects such as soil, water and
different types of forest biota by semiconductor HP Ge detector (Fig. 1).
Fig. 1. Block-diagram of the installation with HP Ge detector.
In order to study radioactivity gamma-spectra as in Fig. 2 should be analyzed.
The spectra in which radionuclides can have the impact on accidents on the environment ought to be found.
3000
counts
2000
Cs-137
1000
0
0
5000
10000
channel
15000
Fig. 2. Gamma-spectrum of the sample.
The studied radionuclides
Artificial radionuclides can track their movement and accumulation in different
environmental objects and can indicate the sources of the pollution. The number
of these radionuclides in the environment determines the degree of pollution of
the study area. In order to characterize the studied accidents radionuclides were
selected from the radionuclides released in this catastrophes.
17
Cesium – one of the substances that have the greatest radiological importance
because of its radioactive isotope 137Cs has a half-decay period, comparable with
the human life; also cesium is very dangerous because of its ability to accumulate
in the human muscle. So for this investigation caesium was selected.
Isotopes of caesium 134Cs and 137Cs is optimally done for the consideration of the
impact of accidents on Fukushima and Chernobyl as they have half-decay periods
close to the time that past from the moment of the accidents (T1/2(134Cs) = 2 years,
T1/2(137Cs) = 30 years) That means that caesium that have another origin has very
low contribution in gamma-spectrum.
The obtained results
The experiments showed that the main contribution in the radioactivity is made
up by caesium 137Cs. Moreover, distribution of 137Cs in this forest ecosystem is not
uniform.
However it is important to note that the radioactivity of 134Cs, which can identify
the Fukushima trace, doesn’t exceed 0.3 Bq/kg in biota, the same data showed in
soil researches. Thus Leningrad region and Saint Petersburg is not appreciably
affected by the contamination from Fukushima.
The average experimental data are shown in Table 1.
compartment
Cs activity
concentration
(Bq/kg)
compartment
Cs activity
concentration
(Bq/kg)
soil
7,8 ±0,1
water
2,0±0,1
needles/leaves
25 ±1
heath
71 ±1
twigs
8,0 ±0,4
fungi
1056 ±3
bark
20 ±1
mosses
198 ±4
cones
14 ±1
herbs
20 ±2
wood mushrooms
17,3 ±0,5
137
137
Table 1. The average experimental data.
Taking into account the average results we arrive to the following conclusion:
• Activity concentration of 137Cs in understorey vegetation is significantly
higher than in the parts of tree. It means that the depth at which plants get their
nutrients affects radioactivity. This confirms the fact that caesium contained in
the top 1-2 cm of soil;
• Radionuclides are distributed throughout the plant, they are absorbed and
move along with the nutrients from the soil and water. So picture of migration
of radionuclides looks like movement of nutrients;
• The amount of caesium in vegetation always higher than in soil. That is, they
absorb cesium from the soil and after the death of the plants again fall into
the ground;
18
• Type and accumulation ability of soil are important factors that affect
absorption of caesium;
• Mushrooms have the highest content of 137Cs in ecosystem – about 1100 Bq/kg.
Consumed radionuclides
As fungi radioactivity of 137Cs is about 1100 Bq/kg, when human consumes this
forest product he is affected by radiation. So if human’s consumption of mushrooms
is around 10 kg per year, he gets a dose that equals 0.2 mSv per year. According
radiation safety standards [3] a maximum permissible annual dose for the population is lower than 1 mSv, it means that the dose from fungi is under permissible
dose, but it contributes significantly into radioactivity.
Conclusions
• The Chernobyl accident significantly affected the pollution of Saint-Petersburg
and Leningrad region since the environment contains 137Cs in large amount;
• The accident at Fukushima did not have a significant impact on the environment
of Leningrad region because radioactivity of 134Cs is low;
• In the investigation radioactivity of different objects of the environment was
measured;
• Certain patterns of the migration of radionuclides in the ecosystem were
identified;
• The dose of radioactivity consumed by population through eating forest
mushrooms was estimated and it showed that the impact is significant.
References
1. Leningrad region. Ecological map. SPb, Discus Media, 2009.
2. Merzlaya A. // Proceedings of the International Student Conference “Science
and Progress”. 12-16 November, 2012. Saint-Petersburg, Russia, pp. 70-74.
3. Sanitary rules and norms 2.6.1.2523-09 "Radiation safety standards (RSS99/2009)".
4. Sapozhnikov U.A., Aliyev R.A., Kalmykov S.A. Radioactivity of the environment. M., Binomial: laboratory of knowledge, 2006.
5. Saharov V.K. Radioecology. -M. 2006.
6. Smith J.T., Beresford N.A. Chernobyl – Catastrophe and Consequences. - UK,
Chichester, Praxis Publishing Ltd. 2005.
19
G. Theoretical, Mathematical
and Computational Physics
Long-range rapidity correlations in a two-component
model
Andronov Evgeny
evgeny.andronov1@gmail.com
Scientific supervisor: Prof. Dr. Vechernin V.V., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University
Introduction
The study of long-range rapidity correlations (LRC) in the processes of the
multiple production in high energy pp and AA collisions can provide us with
information about behavior of the produced high density matter. LRC are the correlations between some observables in two rapidity windows divided by some gap.
One can characterize strength of LRC by the correlation coefficients:
bn − n =
nF
nF nB − nF
nB
2
nB
2
, bpT − n =
nF
pTB
nF pTB − nF
2
ptB
2
n F − nF
n F − nF
where the nF and nB are corresponding multiplicities for the “forward” and “backward” windows,
nB
pTB = ∑ pTBi
i =1
is the event mean transverse momentum in the “backward” window. Symbol ⟨⟩
means averaging over all events.
The soft part of the particle production in high energy pp and AA collisions can
be described in the framework of model with independent identical color strings
[1]. In this type of models it is supposed that several extended color objects [2] are
created between the projectile and the target and then these objects decay, forming
observable particles.
With the growing energy and the number of nucleons of colliding nuclei the
number of emitters grows and it is possible that they will interact with each other
that will lead to the modification of the multiplicity and transverse momentum
distributions.
Results and Discussion
In order to take into account interaction between emitters we propose the model
with two different types of emitters: primary, which were produced immediately
after collision, and secondary, which were produced as a result of interaction of
several primaries. Generalizing formalism developed in [1], it can be shown that for
the multiplicity correlation coefficient in this two-component model we have:
bn − n =
DN1 ⋅ µ F 1 ⋅ µ B1 + c (N1 , N 2 )⋅ (µ B1 ⋅ µ F 2 + µ B 2 ⋅ µ F 1 ) + DN2 ⋅ µ B 2 ⋅ µ F 2
⋅
2
2
N1 ⋅ µ B1 + N 2 ⋅ µ B 2 N1 ⋅ Dµ + N 2 ⋅ Dµ + DN ⋅ µ F 1 + 2 c (N1 , N 2 )⋅ µ F 1 ⋅ µ F 2 + DN ⋅ µ F 2
F1
F2
1
2
N1 ⋅ µ F 1 + N 2 ⋅ µ F 2
22
where N
̅ ̅ 1̅ ̅ and N
̅ ̅ 2̅ ̅ are the mean numbers of primary and secondary emitters, DN and
1
DN are the corresponding variances, the covariance
2
c(N1 , N 2 ) = N1 N 2 − N1 ⋅ N 2 ,
F1 and F2 are the multiplicities in the forward window from all primary/secondary
emitters, B1 and B2 - in the backward window, ̅μ̅F̅ and ̅μF̅ ̅ are the mean values of
1
2
multiplicities produced by one primary/secondary emitter in the forward window,
Dμ and Dμ are the corresponding variances, ̅μ̅B̅ and μ̅B̅ are the mean values of
F1
F2
1
2
multiplicities produced by one primary/secondary emitter in the backward window.
If the properties of the primary and secondary emitters are the same, we come back
to the result obtained in the framework of the model with identical emitters [1]:
DN ⋅ µ F
bn − n =
2
2
N ⋅ Dµ F + µ F ⋅ DN
We propose the simple toy model in order to explain the mechanism of the
emergence of the secondary emitters. Let N be the number of identical emitters
produced immediately after collision. One can consider N as even number due
to assumption, that each pair of emitters corresponds to one cut pomeron of the
Regge-Gribov approach. Then we can assemble N emitters into N/2 pairs and
introduce the probability r of the pair transformation into one secondary emitter.
As a result we will get a binomial distribution for the number of secondary emitters with fixed N:
N 2 − N2
bn − n = PN (N 2 ) = C N2 N 2 ⋅ r N2 ⋅ (1 − r )
Then N1 = N − 2 N 2 and
∞
∑
N1 , N 2 = 0
ω (N1 , N 2 ) =
∞
∞
∑ ∑ P (N )⋅ P (N ) , where ω (N , N ) - the
N = 0 N2 = 0
N
1
2
2
probability to have N1 primary emitters and N2 secondary emitters, P(N) - the
probability to have N primary emitters immediately after collision. One should
also make assumptions on the connection between properties of the primary and
secondary emitters, using idea of the summation of the color fields:
µ F 2 = 2 ⋅ µ F 1 ; µ B 2 = 2 ⋅ µ B1 ; Dµ F 2 = 2 ⋅ Dµ F 1
Then we get:
bn − n =
where
r 2 ⋅ ( 2 − 1) 2 ⋅ (0.5 ⋅ VN − 1) + r ⋅ ( 2 − 1) ⋅ ( 2 − 1 − 2 ⋅ VN ) + VN
r 2 ⋅ ( 2 − 1) 2 ⋅ (0.5 ⋅ VN − 1) + r ⋅ ( 2 − 1) ⋅ ( 2 − 1 − 2 ⋅ VN − W
VN =
DN
,W =
Dµ F 1
2) + VN + W
.
2
µ F1
In Fig. 1 the correlation coefficients for different values of parameters are presented as a function of interaction parameter r. One can see that bn-n can increase
as well as decrease with the growth of the interaction strength. Another important
feature is that with the increase of fluctuations in the number of initial emitters
(i.e. increase of VN) correlations also increase, but even in the absence of fluctuations in the number of initial emitters (VN =0) we still have correlations due to the
interaction of emitters.
23
N
Fig. 1. n-n correlation coefficient as a function of r with W=0.5 and different
values of V = {0; 0.05; 0.5; 1}.
To get expression of the pT-n correlation coefficient we calculate additionally
the following averages:
pTB =
pTB nF =
∑
B1 , B2
∑
B1 , B2
k1 B1 + k2 B2
B1 + B2
k1 B1 + k2 B2
B1 + B2
∑ ω (N , N )P (B )P (B )
1
N1 , N 2
2
∑ ω (N , N )(N µ
1
2
1
F1
N1
1
N2
2
)
+ N 2 µ F 2 PN1 (B1 )PN2 (B2 )
where k̅ 1̅ and k̅ 2̅ are the mean transverse momentum in the backward window from
one primary/secondary emitter, which are connected by ̅k̅2 =21/4 ̅k̅1 .
One can see that we have to average some fractions. It is not possible to do
it explicitly, and we have to make an assumption about denominator in these
formulae:
B1 + B2 ≈ N1 ⋅ µ B1 + N 2 ⋅ µ B 2
Then we get:
34
(1 − r + r 2 )
=
2
bpT − n

N 1 − r + r 21 4 − 1 − r + r

1 − r + r 21 4 r 2 ⋅
(
( 2 − 1) ⋅ (0.5 ⋅V
2
N
− 1)+ r ⋅
)
2 ⋅ ∑ P (N ) PN (N 2 )
N , N2
⋅ 2 −1−
( 2 − 1)(
N − 2N2 + 2 N2 

N − 2N2 + 2N2 
2 ⋅ VN − W
)
2 + VN + W
Here, we still have the sum over N and N2. The calculation of this sum was
performed numerically for the poissonian distribution P(N) with N
̅ ̅ =50 and for
the fixed number of primary emitters N=50. In Fig. 2 results of these calculations
are presented.
One can see that in both cases only negative pT-n correlations take place in
the framework of this simple model. With the event growth of the portion of the
secondary emitters multiplicity decreases while transverse momentum increases,
leading to the negative pT-n correlation. Also, one can note that assumptions about
24
averaging of fractions are
reasonable, because at r=0
and r=1 (configuration with
only one type of emitters)
pT-n correlations vanish.
Negative pT-n correlations have already been observed in experiment [3] in
PbPb collisions as well as
positive. The preliminary results on the NA61 BeBe collisions data show the presence
of negative pT-n correlations.
We see that we need further Fig. 2. pT-n correlation coefficient as a function of r
modification of the model to with W=0.5 and different values of V={0; 2} .
describe this behaviour. One of the possible extensions of the model is the replacement of the constant interaction parameter r with the function
1
r (N ) =
−α (N −β )
1+ e
because it seems reasonable to suppose that with the growth of the number of
emitters in transverse plane the probability to interact should also grow. But with
the function r(N) we can not perform the analytical calculations of the correlation
coefficients till the end. The Monte Carlo simulations algorithm was developed
and tested for constant r, giving values of correlation coefficient consistent with
analytical calculations.
Conclusion
We can make the following conclusions:
a) Explicit formula for bn-n was obtained in the framework of the model with
independent emitters. It was shown that with the inclusion of the interaction by the
simple toy model n-n correlations could increase as well as decrease.
b) Numerical calculations of bpT-n were performed. At present only negative
pT-n correlations were obtained.
c) The possible extension of the model regarding the interaction parameter
was proposed. The Monte Carlo simulations algorithm needed for this extension
was developed.
References
1. Vechernin V.V. // Proc. of Baldin ISHEPP XX, Volume II, 10 (2010);
arxiv:1012.0241v1 [hep-ph].
2. Sicking on behalf of the ALICE Collaboration // CERN LHC Seminar “Multiple
Parton Interactions in ALICE” 05-03-13 https://indico.cern.ch/getFile.py/access?r
esld=0&materialld=slides&confld=238569.
3. Feofilov G.A., Kolevatov R.S., Kondratiev V.P., Naumenko P.A., Vechernin V.V.
// Proc. of Baldin ISHEPP XVII, Volume 1, 222 (2005).
25
Renormalization group analysis of the inertial-range
behaviour of a passive vector field in a random shear
flow
Gulitskiy Nikolay
ngulitskiy@gmail.com
Scientific supervisor: Prof. Dr. Antonov N.V., Department of High
Energy Physics and Elementary Particles, Faculty of Physics,
Saint-Petersburg State University
Introduction
The turbulent advection of a passive scalar impurity has recently attracted
much attention. Having important practical significance in itself, the problem of
passive advection became a cornerstone for studying developed hydrodynamic
turbulence as a whole. The most significant progress in this direction was achieved
for the model proposed by Kraichnan: the infinite set of anomalous exponents
was derived for the first time based on a microscopic model in the framework of
a regular perturbation theory.
In the original Kraichnan model [1], the velocity Vi(x), is chosen to be Gaussian
with a zero correlation time, statistically isotropic and incompressible, with a
pairwise correlation function
VV ∝ δ (t − t ')k − d −ε ,
where k is the wave number, d is the space dimension, and ε is an arbitrary exponent
with the most realistic “Kolmogorov” value 4/3.
In this paper we aim at solving a problem of the vector passive field ϴ, advected
by the velocity field with the prescribed
 
direction
 n:
v (t , x ) = n ⋅ v(t , x⊥ ).
The stochastic equation for advection of the passive field is
∂t θi + ∂ k (vk θi − A0 vi θ k ) + ∂P = ν0 ∆θi + f i ,
which combines as special cases MHD (A0=1), linearization of the Navier-Stokes
equation around the rapid-change velocity field (A0= -1), etc. Here ϴ is the passive
vector field, Δ is the Laplace operator, ν0 is the viscosity coefficient, P is the pressure
and fi is the random foreign force with
 zero mean and preassigned correlator
f i (t , x ) f k (t ', x ') = δ (t − t ')Cik (r / L).
In the above expression r=x-x', the parameter L is an external turbulence scale
and connected to the stirring, and Cik is a dimensionless function which is finite as
L→0 and rapidly decays as L→∞.
In the real problem, the velocity v(t, x) satisfies the Navier-Stokes equation,
probably with additional terms that describe the feedback of the advected field ϴ.
But here we will consider a simplified model, where the statistics of v(t, x) is given:
it is a Gaussian field withzero mean
 and correlation
 function

vi (t , x )vk (t ', x ') = ni nk v(t , x⊥ )v(t ', x⊥ ')
where
26

dk ik ( x − x ')


v(t , x⊥ )v(t ', x⊥ ') = δ (t − t ') ⋅ ∫
e
Dv (k )
(2π) d
with some function Dv(k), for which we choose
1
Dv (k ) = 2πδ (k parallel ) D0
.
k⊥
Here δ is delta-function, d is the dimensionality of x-space, 1/m is another turbulence scale, connected to the attenuation, the exponent ε plays the role of the RG
expansion parameter and D0>0 is an amplitude factor.
Field theoretic formulation and renormalization
According to the general theorem [2], this full-scale stochastic problem is
equivalent to the field theoretic model of the doubled set of fields with action
functional
S (Φ ) = θ 'k  −∂t θ k − (vi ∂ i )θ k + A0 (θi ∂ i )vk + ν0 (∂ 2⊥ + f 0 ∂ 2par )θ k  +
d −1+ ε
1
1
+ θ 'i D f θ k − vi Dv−1vk .
2
2
This action functional provides us with the triple vertex Vabc and two propagators:

Pik (k )
'
,
θi θ k 0 =
=
2
)
−iω + ν0 (k⊥2 + f 0 k parallel
θi θ k
=
=
Cik (k )
.
2
) 
ω +  ν0 (k⊥2 + f 0 k parallel
The analysis of UV divergences is based on the analysis of canonical dimensions of the 1-irreducible Green functions (for detailed discussion see [3, 4]). The
total canonical “reduced” dimension of an arbitrary 1-irreducible Green function
Γ is given by the relation
d Γ ' = (d + 2)(1 − N θ ') − N v .
Superficial UV divergences, whose removal requires counterterms, can be presented only in those functions Γ for which the “reduced index of divergence” dˊΓ is
a nonnegative integer. Therefore we are left with the only superficially divergent
function, for which Dyson
like
 equation looks


Γ 2 = −iω ⋅δ12 + ν0 k⊥2 ⋅δ12 + ν0 f 0 ⋅ ( pn ) 2 ⋅δ12 + ν0 f 0 u0 ⋅ ( pn ) 2 ⋅ n1n2 − Σ ,
where Σ is self-energy operator, and in diagram notation it is
0
Σ = 2
+ …,
where three dots denotes 2--, 3-- and other N--loop diagrams. But, fortunately, in
our model all N--loop diagrams contains closed cycles of retarded propagators
27
and therefore vanish, i. e. self-energy operator is expressed via one-loop diagram
precisely.
From the analysis of Dyson equation it follows, that
Z ν = 1, Z A = 1, f 0 = fZ f , u0 = uZ u , g 0 = g µ ε Z g , Z g = Z −f 1 ,
and for RG-functions in a fixed point one can obtain, that
g* =
2(d − 1)
( A − 1) 2
, ..... u * =
.
d −2+ A
d −2+ A
Therefore, the system possesses fixed point u*, g* only if g*>0, i. e.
d − 2 + A > 0.
This fact implies that correlation functions of the model in the IR region
Λr>>1, mr ≈1 exhibit scaling behavior; the corresponding critical dimensions can
be calculated as series in ε.
From now on, we shall consider composite operators
FN , p , m = (θi θi ) p (ns θ s ) 2 m , ... N = 2( p + m).
They are renormalized multiplicatively, and the renormalization constants ZNp are
determined by the requirement that the 1-irreducible correlation function
−1
FNpR ( x)θ( x1 )...θ( xn )
= Z Np
F Np ( x)θ( x1 )...θ( xn )
1− ir
be UV finite in renormalized theory, i.e., have no poles in ε when expressed in
renormalized variables.
Fortunately, unlike the scalar model with strong anisotropy [5], in this model
contribution to the generating functional is given only by one-loop diagram
The internal structure for it is 2
A
m− ε

I cdab =
⋅ Pcd (n ) ⋅ na nb ⋅ Cd −1 ⋅
.
2 ν(d − 1)
ε
Let F={Fi} be a closed set of operators with the same quantity of fields ϴ, i. e.
with the same number N, which mix only with each other in renormalization. The
renormalization matrix ZF={Zik} and the matrix of anomalous dimensions γF={γik}
for this set are given by
Fi = ∑ Z ik FkR , ... γ F = Z F−1 Dµ Z F .
k
From hard, but direct calculations one can obtain, that
A2
γ *N , p +1 = −
⋅ Cd ⋅ 2m(2m − 1) ⋅ ε;
2(d − 2 + A)
2
A
γ *N , p = −
⋅ Cd ⋅ (2 p + 8 pm − 2m(2m − 1)) ⋅ ε;
2(d − 2 + A)
γ *N , p −1 = −
A2
⋅ Cd ⋅ (4 p ( p − 1) − 2 p − 8 pm) ⋅ ε;
2(d − 2 + A)
28
γ *N , p − 2 = −
A2
⋅ Cd ⋅ (−4 p ( p − 1)) ⋅ ε.
2(d − 2 + A)
Therefore the critical dimensions matrixes for operator FNp have the form
∆ Np , Np ' = −2( p + m) ⋅δ pp ' + γ *Np , Np ' ,
where -2(p+m) is its canonical dimension, δpp' is Kronecker's delta-symbol and
γ*Np,Np' is the value of the anomalous dimension at a critical point. This matrix is
degenerate with all eigenvalues are equal to its canonical dimension:
λ1 = ... = λ n = −2( p + m).
This fact has deep physical meaning: it states, that asymptotic behaviour is governed
not only by power law, but it is a combination of power and logarithm.
Operator product expansion and asymptotic behaviour
The basic RG equation for a multiplicatively renormalizable quantity (correlation function, composite operator, etc) is the consequence of operating with
differential operation μ d/dμ to a relation F=ZFFR and it has the form
[DRG + γ F ]FR = 0.
The solution of it for pair correlation function
G = FN1 , p1 FN 2 , p 2
is
(
)
G ∝ νdG ⋅ r − ( N1 + N2 ) ⋅ P( N1 + N 2)/ 2 [ln Λr ]⋅ Φ Mr , mr , fr ε ,
where Px is a polynomial function power x and Φ is the function of three dimensionless arguments. The presence of the logarithm function in the above formulae
is unexpected and very interesting feature of this model. Both scalar anisotropic
model and vector magneto-hydrodynamical model [6] doesn’t exist this peculiarity.
This representation describes the behavior of the correlation functions for Λr>>1
and any fixed values of Mr and mr. The inertial range corresponds to the additional
condition Mr<<1. The form of the function Φ is not determined by the RG equations themselves; in analogy with the theory of critical phenomena, its behavior for
Mr→0 is studied using the well-known Wilson operator product expansion (OPE).
Note, that we assume, that function Φ is finite as its third argument is near zero.
According to the OPE, the equal-time product F1(x)F2(x’) of two renormalized
operators for x=(x+x’)/2=const and r=x-x’→0 has the representation


F1 ( x) F2 ( x ') = ∑ CF (r ) ⋅ F (t , x ),
ω
F
where the functions CF are coefficients regular in M2 and F are all possible renormalized local composite operators allowed by symmetry.
Being applied to our model, together RG and OPE approaches give us the desired
asymptotic behaviour of the pair correlation function in the inertial range:
( )
G ∝ νdG ⋅ m − ( N1 + N2 ) ⋅ [ln Λr ] 1 2 ⋅ [ln Mr ] 1 2 ⋅ Ψ fr ε ,
where Ψ is some finite function as its argument is near zero.
This asymptotic has a very interesting property: unlike other models, it logarithmically increases as r runs high. The reason for that might be that our approach
29
ω
( N + N )/ 2
( N + N )/ 2
works well only in the inertial interval, namely only if 1/Λ <<r<<1/m. Thus if r is
too large, this formulae does not work and the correlator naturally decreases.
Conclusion
In this paper the inertial range asymptotic behaviour of the correlation function
G of two composite operators was investigated using renormalization group approach and operator product expansion. The degeneracy of the critical dimension
matrix is that reason, for which it exhibits anomalous behavior: it obtain not power,
but logarithmic relation as the distance between two points.
References
1. Kraichnan R.H. // Phys. Fluids 11, 945 (1968).
2. Vasil'ev A.N., The Field Theoretic Renormalization Group in Critical Behavior
Theory and Stochastic Dynamics. - St. Petersburg Institute of Nuclear Physics, St.
Petersburg (1998).
3. Adzhemyan L.Ts., Antonov N.V. // Phys. Rev. E 58, p. 7381 (1998).
4. Adzhemyan L.Ts., Antonov N.V., Vasil'ev A.N. // Phys. Rev. E 58, p. 1823
(1998); chao-dyn/9801033.
5. Antonov N.V., Malyshev A.V. //Journal of Statistical Physics 146/1, pp. 33-55
(2012).
6. Antonov N.V., Gulitskiy N.M. // Phys. Rev. E 85, p. 065301(R) (2012).
30
Splint in case of special embeddings
Kakin Polina
megachi@yandex.ru
Scientific supervisor: Prof. Dr. Ioffe M.V., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University
Introduction
The term "splint" was introduced by D. Richter in [1] where the classification
of splints for classical root systems was obtained. It was shown in [2] that not only
splint had tight connections with the injection fan construction but that it could
be used as a tool to study reduction properties of g-modules with respect to a corresponding subalgebra and, in particular, that it allowed to drastically simplify
calculation of branching coefficients for the reduced modules.
After such a discovery the next logical step would be to try to generalize the
concept of splint. Previously splint construction was applied only in case of regular
embeddings i.e. in a situation where both embeddings splintering main root system
were regular. The aim of the present work is to generalize splint on case of special
embeddings when one of the embeddings is special and therefore a corresponding
root system can not be identified with a subset of a main root system.
We study some classical root systems and show the way such a generalization
can be performed. The fact that root system splinters even in an unlikely situation
like this may be pointing to the existence of yet unknown properties of singular
elements of representations.
Results and Discussion
A root system is said to be splintered when there are two root systems that can
be regularly embedded into it and when it is a disjoint union of the images of these
embedded root systems. These embedded root systems are called stems and their
ranks should be no bigger than the rank of the main root system. If the algebra
corresponding to one of the stems is a subalgebra of the algebra of the main root
system then in terms of injection fan it means that injection fan of this subalgebra
is equal to the star of other stem. Thus splint becomes a tool to study reduction
properties of g-modules. When one of the embeddings is special though, one can
no longer talk about a disjoint union. How then splint can be defined? It turns out
that in case of special embeddings so(2n-1)→so(2n) projection of the main root
system made in such a way that image of specially embedded root system is part
of the projection of the main root system can be splintered and allows to perform
reduction of representation of so(2n) on subrepresentations of so(2n-1) despite the
projection not being a classical root system.
31
Fig. 1. From left to right: projection of the root system of so(6), projection of
the image of specially embedded so(5), projection of the rest of the root system of so(6).
In the Fig. 1 result of such an approach can be seen. Specially embedded root
system of so(5) can not be identified with any subset of the root system of so(6)
but after projection the new “root system” splinters into so(5) and so(4). Let us
look at the projection of singular element of module of so(6) with Dynkin num-
Fig. 2. Projection of the singular element of module of so(6) with Dynkin numbers [1,0,0] is depicted by black dots (singular multiplicities are near the dots),
fundamental Weyl chamber of so(5) is depicvted by grey area, weight diagram
of so(4) is depicted by grey dots (weight multiplicities are near the dots).
bers [1,0,0], we can see that by applying injection fan of the projection of so(5)
to it we get weight diagram of so(4) in thefundamental Weyl chamber of so(5) in
complete accordance with the Dynkin numbers of module of so(6). This injection
fan turns out to be equal to the star of so(4). In other words, projection of the root
system of so(6) indeed splinters. This phenomenon occurs for every module of
so(6). Moreover projection of root system of any algebra so(2n) consists of root
system of so(2n-1) and set of vectors {ei, 1 ≤ i ≤ n-1} (where {ei, 1 ≤ i ≤ n} is
32
orthonormal basis) which ensures that for special embeddings so(2n-1)→so(2n)
there is always splint which is also consistent with the results of reduction by
method of Gelfund-Zeitlin basis.
Unusual root system that appeared in the case described above is one of those
root systems for which there is no developed representation theory. Nevertheless,
we were able to use the concept of splint. However it is not always possible. For
example, in case of special embedding sl(2)→sl(3) projection of the root system of
sl(3) splinters but one of the stems turns out to be another unusual root system with
collinear roots and as such results of splinting can not be verified. Yet a concept of
splint may shed a light on the representation theory of such root systems.
Conclusion
It was found out that it is possible to apply concept of splint to special embeddings using projection of the root system.
It was also discovered that in case of special embeddings so(2n-1)→so(2n) splint
yields the same results as the reduction by the method of Gelfund-Zeitlin basis.
A phenomenon of splint is yet mostly unexplored but it might present unexpected possibilities for discovering new properties of singular elements, root systems
with collinear roots and special embeddings which otherwise are hard to study.
As splint allows us to drastically simplify calculations of branching coefficients
and can be generalized on special embeddings it becomes a useful instrument of
representation theory.
References
1. Richter D. // J. Geom. 103 (2012), pp. 103-117.
2. Lyakhovsky V.D., Nazarov A.A. // Arxiv preprint arXiv:1111.6787v2 (2011).
33
Critical behavior of the O(n)-φ4 model with an
antisymmetric tensor order parameter
Lebedev Nikita
moisey2029@rambler.ru
Scientific supervisor: Prof. Dr. Antonov N.V., Department of High
Energy and Elementary Particle Physics, Faculty of Physics, SaintPetersburg State University
Introduction
Numerous physical systems reveal interesting singular behavior in the vicinity
of their critical points. Their thermodynamical and correlation functions exhibit
scaling behavior with universal critical dimensions: according to general belief,
they depend only on a few global characteristics of the system, such as symmetry
or dimension. The powerful and quantitative theory of the critical behavior is
provided by the field theoretic renormalization group (RG). In the RG approach,
possible types of critical behavior are associated with infrared (IR) attractive fixed
points of renormalizable field theoretic models.
In the present work we apply the field theoretic RG to the O(n)-symmetric φ4
model of the real n-th rank tensor order parameter. This model can be relevant in
the analysis of transitions between the nematic, cholesteric and blue phases in liquid
crystals, transitions to ferroelastic state in solids and transitions to the superconductive state in systems with higher spins. In order to simplify and to sharpen the
problem, we consider the case of a purely antisymmetric tensor. In comparison to
the general nth rank tensor case, this reduces the model to the two-charge problem,
which makes the results more visible.
Results and Discussion
We study a model of a real antisymmetric nth rank tensor field φ = φik (so that
φ = –φik and i, k = 1,…, n) in the Euclidean d-dimensional x space. The action
functional has the form:
with the free part
and the interaction term with the two independent quartic structures
The action is invariant with respect to the transformation φ → OφO†, where
O ∈ O(n) is an nth rank orthogonal matrix (note that the antisymmetry property is
preserved by this transformation).
The stability of the model requires the interaction term to be negative for all
values of φ. One can check that the condition V(φ) < 0 imposes the following
restrictions on the coupling constants:
34
for even values of n and
for n odd.
For n = 2 and n = 3 the model reduces to the well-known cases: the single
component φ4 model and the O(3)-invariant vector model, respectively.
The Feynman diagrammatic techniques for the model are derived in a standard
fashion; see e.g. [1-3]. In the momentum (Fourier) representation the bare propagator, determined by the free action, has the form
where p = |p| is the wave number. The tensor
built from the Kronecker δ symbols, is antisymmetric with respect to the transpositions of its indices i ↔ k and l ↔ m, and symmetric with respect to the transposition of the pairs ik ↔ lm. It plays the part of the unit operation on the space of
antisymmetric tensors in the sense that
and
The interactions V1,2(φ) correspond to the quartic vertices with the vertex
factors
where the tensors
are defined such that
Thus any diagram of our model is represented as a product of two factors: the
corresponding diagram for the single-component φ4 model with the corresponding
symmetry coefficient and the additional n-dependent factor stemming from the
contractions of the tensors in the propagators and vertices.
The analysis of renormalizability of the model is very similar to the case of the
single-component φ4 model; see e.g. [1-3]. The model is logarithmic (the coupling
constants are dimensionless) for d = 4. In the dimensional regularization, the UV
divergences have the form of the poles in ε = 4 − d, the deviation of the dimension
of space from its upper critical value d = 4. Standard analysis, based on the dimensionality and symmetry considerations, shows that superficial UV divergences,
35
whose elimination requires counterterms, are present only in the 1-irreducible
Green functions ‹φφ› and ‹φφφφ›. The required counterterms have the same forms
as the terms already present in the action and can therefore be reproduced by the
multiplicative renormalization of the field and the model parameters.
The corresponding renormalized action has the form
This expression can be reproduced by the multiplicative renormalization of
the field
and the parameters
so that
We use the minimal subtraction (MS) scheme, where all renormalization constants have the forms ‘1+ only poles in ε,’
with the coefficients depending only on the completely dimensionless renormalized
couplings. The explicit one-loop calculation gives
in order to simplify the coefficients, here and below we pass to the new couplings:
g1,2 → g1,2/(8π2).
Like in the ordinary ϕ4 model, the nontrivial contributions to the constant Z1
appear only in the two-loop order:
In our model the RG equation for the renormalized n-point connected Green
functions has the form:
The RG functions (β-functions for the coupling constants and anomalous dimensions γ) are defined by the relations
from this equations and the explicit form of the renormalization constants we
obtain:
36
Possible asymptotic regimes of a renormalizable field theoretic model are
determined by the asymptotic behaviour of the system of ordinary differential
equations for the so-called invariant coupling constants
Here s = k/μ is a non-dimensionalized momentum. As a rule, the IR (s → 0) and
UV (s→∞) behavior of the Green functions is determined by fixed points of the
this system
The type of fixed point is determined by the matrix
Analysis of the one-loop expressions reveals the following fixed points:
1) The Gaussian (free) fixed point with both vanishing coordinates, UV
attractive (IR repulsive) for all n with both eigenvalues ω = −ε.
2)The point
For all n ≥ 4 it lies in the physical region, but is a saddle point: the
eigenvalues
are real and opposite in sign.
3)Two fully non-trivial points with both non-vanishing coordinates:
For all n ≥ 5, however, these points are complex and thus cannot be reached
by the RG flow with real initial data. In case n=4 the first point is IR attractive
with the eigenvalues ω1 = ε, ω2 = ε/17, while the second one is a saddle
point with ω1 = ε, ω2 = −ε/11.
The existence of an IR attractive fixed point implies the existence of scaling
behavior for all the Green functions, described by the two main independent critical exponents [1-3]
where γ with a star means the values of anomalous dimensions at the fixed point
in question. For the saddle fixed point one obtains:
37
and for IR attractive fixed point (n=4):
Fig. 1. RG flows for n = 4.
Conclusion
We conclude that the IR behavior in our model for n = 4 is non-universal in the
sense that it depends on the choice of the couplings. If they belong to the basin of
attraction for the IR point (in particular, this implies g2 < 0), the phase transition
is of the second order (and thus the scaling regime takes place). Otherwise the
RG flows pass outside the stability region. For n > 4, there are no attractive fixed
points and only the second possibility can be realized. This means that the account
of fluctuations changes the nature of the phase transition from the second-order
type (suggested by the mean-field theory) to the first-order type. For more detailed
analysis of model see [4].
References
1. Zinn-Justin J. Quantum Field Theory and Critical Phenomena. Oxford:
Clarendon, 1989.
2. Vasil’ev A.N. The Field Theoretic Renormalization Group in Critical Behavior
Theory and Stochastic Dynamics. Boca Raton, FL: Chapman and Hall, 2004.
3. Kleinert H., Schulte-Frohlinde V. Critical Properties of φ4-Theories.- Singapore:
World Scientific, 2000.
4. Antonov N.V. et al. // J. Phys. A: Math. Theor. 46, 405002, 2013.
38
Spectral properties of quasi-periodic Schrodinger
equations
Martemyanov Andrey
andrew.martemyanov@yandex.ru
Scientific supervisor: Prof. Dr. Fedotov A.A., Department of
Mathematical Physics, Faculty of Physics, Saint-Petersburg State
University
Introduction
In this work, we study the family of Schrodinger equations:
d2
− 2 ψ ( x, z ) + (V ( x) + αU ( x − z ))ψ ( x, z ) = E ψ ( x, z ), x ∈ �
dx
Functions V(.) ∈ L2(0,1) and U(.) ∈ L2(0,ω) are real-valued and periodic,
the period of V equals one, the period of U(.) equals ω. Here, ω is an irrational
number. The periods of these functions are incommensurable, and this equation
is quasi-periodic. Furthermore, we assume that U(.) is analytical in the set {x∈ℂ:
|Im x|<r, r>0}.
The E is real and called a “spectral” parameter. The parameter z indices the
equations of the family, z ∈ (0,ω). The α is the coupling constant. We assume that
it is positive.
The quasi-periodic Schrodinger operator is a model for electrons in a crystal
placed in an external periodic electric field. Usually, the external field is much
weaker than the internal field of the crystal. This allows us to assume that the
coupling constant α is small.
In this work we consider a small quasi-periodic perturbation of a periodic
Schrodinger operator. The principal purpose of the present work is to study the
spectrum of the perturbed one. In particular, we prove that the main part of the
absolutely continuous spectrum of the unperturbed Schrodinger operator remains
intact. Moreover, we also suggest a procedure for constructing generalized eigenfunctions of the quasi-periodic operator for some set of parameters. It is based on
works of V. Buslaev, A. Fedotov and F. Klopp.
Theorem: Let Ω be a closed interval of the real axis contained on a spectral
band of the periodic Schrödinger equation
 d2

 − 2 + V ( x)  ψ 0 ( x) = E ψ 0 ( x), x ∈ �
 dx

Then, for sufficiently small α and some diophantine ω, there is a set Ω∞ÌΩ such
that
1. Ω∞ is contained in the absolutely continuous spectrum of the quasi-periodic
Schrödinger operator for any z∈(0,ω).
2. Lebesgue measure of Ω\ Ω∞ tends to zero as α→0.
39
3. the corresponding eigenfunctions are of Bloch-Floquet type in the sense of
Dinaburg and Sinai [4].
Periodic operator
First, recall some facts about periodic Schrodinger operator
d2
− 2 ψ 0 ( x) + V ( x)ψ 0 ( x) = E ψ 0 ( x), x ∈ �.
dx
As the function V(.) is periodic, this operator is invariant with respect to
translation by 1. Consequently, if functions φ1,2(.) forms a basis for the space of
solutions
 ϕ1 ( x + 1) 
 ϕ1 ( x) 
 ϕ ( x + 1) = M 0  ϕ ( x) ,
2
2
where M0 is matrix independent of x. M0 is called monodromy matrix. It is well
known that the spectral analysis of a periodic Schrodinger equation is reduced to
calculation of the eigenvalues and eigenfunctions of the monodromy matrix M0.
The spectrum of the periodic Schrodinger operator has a band structure, i.e.,
it consists of a countable set of non-overlapping intervals of the real axes that can
accumulate only to +∞. For all values of the spectral parameter, except the ends
of the spectral bands, there exists a basis consisting of Bloch solutions, that are
of the form
ik ( E ) x
ψ ± ( x) = e
p ( x),
0
±
where p±(x) is a 1-periodic function, k(E) is a multivalued analytical function of
E, called Bloch quasi momentum.
It should be noted that when E belongs to the spectrum of the operator the Bloch
functions are continuous, bounded and differ by complex conjugation.
Method of monodromization
Now, come back to the quasi-periodic Schrodinger equation. We analyze it by
the method of monodromization, which arised when trying to extend the notion
of monodromy matrix to the quasi-periodic case [1].
Note, that it is always possible to choose a basis ψ1,2 for the solution space
which has the properties: ψ1,2(x, .) are ω-periodic by z and wronskian of ψ1,2 is
independent of z. Such a basis is called consistent.
Coefficients of the equation are invariant with respect to a simultaneous translation of x and z by 1. For a consistent basis ψ1,2(.)
 ψ1 ( x + 1, z + 1) 
 ψ 1 ( x, z ) 
 ψ ( x + 1, z + 1) = M ( z )  ψ ( x, z )
2
2
The M(z) is called a monodromy matrix of the family of quasi-periodic Schrodinger
equations. It is easy to show that M(z) is ω-periodic and
det M(z)≡1.
Now the spectral analysis leads to a study of solutions of the one-dimentional difference quasi-periodic equation
χ m +1 ( z ) = M ( z + m)χ m ( z ), χ : � → �2 , z ∈ �,
40
which is called a monodromy equation. Behaviour of its solutions for m→±∞ copies
behavior of solutions of the input differential equation for x→∞.
In this talk, analyzing the monodromy equation, we get information on the
input one.
KAM-theory
Consider the equation
  eik

0 
γ ( z + h) =  
+ αA( z ) γ ( z ), z ∈ �
− ik 
 0 e 

Here h∈(0,1), A(.) is a matrix function, α is a small constant. We assume that
  eik
det  
 0

0 
+ αA( z ) ≡ 1
− ik 
e 

Procedure of constructing [1] bounded solutions of this equation is based on
the following idea: coefficients of the equation are independent of z in the leading
order. Only A depends on z, and one looks for a transformation of this equation
which reduces it to an equation with coefficients independent of z.
For this, we construct its solution in the form:
γ ( z ) = (I + v1 ( z ) )γ 1 ( z )
where v1(.) is a “small” function. After substitution of this expression into the
input equation, we get a similar equation for γ1(.), but the small term depending
on z becomes smaller. By repeating this procedure infinitely many times we reach
the desired aim.
When carrying out this procedure, one encounters the problem of "small denominators", which leads to certain conditions on ω and E.
Calculations
To prove the theorem, we need to calculate a monodromy matrix. Consider
the integral equation
ψ + ( x, z ) = ψ 0+ ( x) −
α
W
x
∫ (ψ
+
0
)
( x)ψ 0− (t ) − ψ 0+ (t )ψ 0− ( x) U (t − z )ψ + (t )dt ,
0
W is the wronskian of the Bloch solutions ψ0±(.). It is easy to show that the equation
has a unique solution which satisfies to the quasi-periodic Schrodinger equation.
We can use Neumann series for bounded operators to get the asymptotics of its
solutions for α→0
x
α
ψ + ( x, z ) = ψ 0+ ( x) − ∫ ψ 0+ ( x)ψ 0− (t ) − ψ 0+ (t )ψ 0− ( x) ψ 0+ (t )dt + O(α 2 )
W 0
(
)
Similarly, we construct ψ-(.). Functions ψ±(.) form a consistent basis for the space of
solutions of the original equation. Using the equality from section devoted to method
of monodromization, we also get the asymptotics for the monodromy matrix
 eik
M ( z) = 
 0
0 
+ αM ( z ) + O(α 2 ).
e − ik 
41
Obviously, the leading term is independent of z. Therefore, the monodromy equation can be written in the form
  eik
χ m +1 ( z ) =  
 0


0 
+ α A( z + m)  χ m ( z ).
− ik 

e 

Together with the previous equation, consider the equation
  eik
χ( z + 1) =  
 0

0 
+ αA( z ) χ( z ).
− ik 
e 

In term of its solution, one can construct a solution of the monodromy equation
by the formula:
χ( z + m) = χ m ( z )
So, we got bounded solutions of the monodromy equation. As mentioned
earlier, their behavior repeats the behavior of solutions of the input Schrodinger
equation for x→±∞. In particular, knowing a bounded solution of the monodromy
equation it is possible to get a bounded solution of the differential equation. To
complete the proof of our main result, one has only to refer to Ishii-Pastur-Kotani's
theorem [2].
References
1. Fedotov A. and Klopp F. // Communications in Mathematical Physics, 227:192, 2002.
2. Pastur L., Figotin A. Spectra of Random and Almost-Periodic Operators.
Springer Verlag, Berlin, 1992.
3. Buslaev V., Fedotov A. // Advances in Theoretical and Mathematical Physics,
5(6):1105-1168, 2001.
4. Dinaburg E.I., Sinai Ya.G. // Funkts. Anal. Prilozh., 9:4 (1975), 8–21.
42
Calculation of self-energy diagrams in the φ4 theory with
help of recurrence relations
Pismenskii Artem
artem5085@mail.ru
Scientific supervisor: Prof. Dr. Pis’mak Yu.M., Department of
High Energy and Elementary Particle Physics, Faculty of Physics,
Saint-Petersburg State University
We consider the quantum field theory in the Euclidian space with the following Lagrangian:
1
λ
2
 = (∂ϕ ) + ϕ 4
2
4!
φ is a scalar field, λ is a coupling constant, λ>0.
The aim of the present work is to compute the self-energy operator in 4-loop
approximation in the logarithmic dimension.
The self-energy operator of the φ4 theory is presented as an infinite sum of
1-particle-irreducible diagrams. That in 4-loop approximation is the following:
We denote these diagrams as Σ1, Σ2, Σ3, Σ4, Σ5 and Σ6, respectively.
We perform calculations in the momentum representation:
There are two basic formulas:
Here we use the following notation: d is a space dimension, ω = d / 2,
H (α ) =
Γ (ω − α )
,
Γ (α )
H (α1 , a2 , α 3 , …) = H (α1 )H (α 2 )H (α 3 )…
Using these basic formulas one can calculate the graphs Σ1 till Σ4.
43
Σ1 =
Σ2 =
Σ3 =
Σ4 =
1
(4π)d
H (1,1,1, 3ω − 3)
1
p 2(3− d )
H (1,1, d − 2) H (1, 4 − d , 2d − 5)
λ3
2
(4π)
3ω
1
p
H (1,1, d − 2) H (1, 6 − 3ω,5ω − 7)
2(5 − 3ω )
λ4
3
(4π)
2d
1
λ2
(4π)2 d
p
H (1,1,1, 3ω − 3) H (1,1,5 − d ,5ω − 7)
2(7 − 2 d )
λ4
p 2(7 − 2 d )
The diagrams Σ5 and Σ6 are not calculated in this way. To compute them we
used recurrence relations.
We introduced two auxiliary graphs:
with arbitrary α.
The diagrams Σ5 and Σ6 are expressed through the graphs γ1 and γ2, respectively,
namely:
1
λ4
Σ5 =
H (1,1, d − 2)H (1, 6 − 3ω, 5ω − 7) γ 1 (2 − ω ) 2(7 − 2 d )
d
p
(4π)
Σ6 =
1
H (1,1, d − 2) γ 2 (2 − ω )
2
λ4
p
(4π)
We received two recurrence relations: one connects γ1(α) with γ1(α–1) and
another does γ2(α) with γ2(α–1) with arbitrary α.
For obtaining the first recurrence relation we used the following two formulas:
d
2(7 − 2 d )
The sign “plus” at a line means the increasing of the index of the line by unity,
the sign “minus” – decreasing by unity. Remark: these formulas are written for
graphs in the coordinate representation, but they are true in the momentum representation also.
Applying the first formula to γ1(α–1) and the second to γ1(������������������
����������������
) we got the following recurrence relation:
44
(α + 2 − d ) γ α − 1 +
( )
(α + 1 − ω) 1
 (3ω − 4 − α )(3 + α − d )
1
1
+
1−
×
d
(α − 1)(ω − α ) 
(4π) (ω − 1 − α ) 
× H (2, α − 1, d − 1 − α,1, 2 + α − ω, 3ω − 3 − α ).
γ 1 (α ) = −
To obtain the recurrence relation for γ2(α) we applied the formula
to γ2(α) and to the graph
We also used the formulas of differentiation:
We applied ∂2 to γ2(α–1) and to the graph
We received a system of 4 equations, from which we got the following recurrence
relation:
γ 2 (α − 1) =
(1 + α − ω)2
1
1
π
γ (α ) +
Γ (ω − 1) ×
(2 + 2α − 3ω )(3 + 2α − 3ω ) 2
(4π)d Γ (α )2 sin (π (α − ω ))
1


22α − d (3α − d )Γ ( + α − ω )Γ (ω − α )
2(6α 2 + α (8 − 6d ) + (d − 2)(d + 1))π 3/ 2 Γ (ω − 1)


2
×
+
2

(2 + α − d )Γ (3ω − 1 − 2α )
 (4 + 4α − 3d )(6 + 4α − 3d )Γ (d − α − 1) sin(π (α − ω ))



The logarithmic dimension for the φ4 theory is d=4. In this dimension all the
diagrams of the self-energy operator are diverged, and we used the dimensional
regularization (d=4–2ε) to calculate them. All the singularities are shown in the
form of poles in the ε.
In the dimension d=4–2ε we have:
Σ1 =
Σ2 =
Σ3 =
1
Γ (1 − ε ) Γ (−1 + 2ε) λ 2
Γ (3 − 3ε)
p 2( −1+ 2 ε )
1
Γ (1 − ε ) Γ (2 − 3ε )Γ (ε ) Γ (−1 + 3ε)
(4π)4− 2 ε
(4π)6−3ε
1
(4π)8− 4 ε
3
5
2
λ3
Γ (2ε )Γ (2 − 2ε ) Γ (3 − 4ε)
p 2( −1+ 3ε )
Γ (1 − ε ) Γ (2 − 4ε )Γ (ε ) Γ (−1 + 4ε)
λ4
2
7
3
Γ (3ε )Γ (2 − 2ε ) Γ (3 − 5ε )
3
45
p 2( −1+ 4 ε )
Σ4 =
Σ5 =
Σ6 =
1
Γ (1 − ε ) Γ (1 − 3ε )Γ (−1 + 2ε )Γ (−1 + 4ε )
λ4
2( −1+ 4 ε )
Γ (1 + 2ε )Γ (3 − 3ε )Γ (3 − 5ε )
p
1
Γ (1 − ε ) Γ (ε )Γ (2 − 4ε )Γ (−1 + 4ε )
λ4
γ 1 (ε ) 2( −1+ 4 ε )
Γ (3 − 5ε )Γ (2 − 2ε )Γ (3ε )
p
1
Γ (1 − ε ) Γ (ε )
(4π)8− 4 ε
(4π)4− 2 ε
(4π)4− 2 ε
5
3
4
2
Γ (2 − 2ε )
2
γ 2 (ε)
λ4
p 2( −1+ 4 ε )
To calculate γ1(ε) and γ2(ε) we used the fact that
Therefore, it holds
γ 1 (1 + ε ) =
γ 2 (1 + ε ) =
1
(6ζ (3) + O (ε ))
1
(6ζ (3) + O (ε ))
(4π)4− 2 ε
(4π)4− 2 ε
Here ζ(x) is the Riemann’s zeta function.
This approximation is sufficient for finding the singular and the finite parts of
Σ5 and Σ6.
Using the recurrence relations we found γ1(ε) and γ2(ε), and then, we calculated
Σ5 and Σ6.
After these calculations we did the renormalization in the minimal subtractions
(MS) scheme.
ϕ = Zϕ ϕ R , λ = µ 2ε Zλ λ R ,
where Zφ and Zλ are renormalization constants, φR λR are renormalized field and
coupling constant, μ is a parameter with the dimension of mass. We made the
R-operation to receive a finite result for the self-energy operator in the dimension
d=4 in terms of the renormalized values.
The result of the application of the R-operation to all the diagrams:
13 + 4τ 2 2
R Σ1 = −
g p
8
167 + 84τ + 12τ 2 3 2
RΣ 2 = −
g p
24
1851 + 1296τ + 336τ 2 + 32τ 3 + 16 ζ (3) 4 2
RΣ 3 = −
g p
64
543 + 184τ + 16τ 2 4 2
RΣ 4 =
g p
128
3333 + 2064τ + 432τ 2 + 32τ 3 + 352 ζ (3) 4 2
RΣ 5 = −
g p
192
46
RΣ 6 = −
Here τ = ln 4π − γ E − 2 ln
783 + 444τ + 96τ 2 + 8τ 3 − 56 ζ(3) 4 2
g p
24
λ
p
, g = R2
µ
16π
The result for the self-energy operator in the MS-scheme is the following:
 13 + 4τ 2 167 + 84τ + 12τ 2 3
g −
g −
Σ = −
48
96

−

6051 + 3734τ + 848τ 2 + 72τ 3 − 36ζ (3) 4
g + … p 2
384

Conclusions
We have found the self-energy operator for the φ4 theory in 4-loop approximation in the logarithmic dimension d=4. Some diagrams are not calculated
explicitly – we cannot express them through a finite product of gamma-functions.
We have computed the result of the application of the R-operation to them using
the dimension regularization (d=4–2ε) and the minimal subtractions scheme of
renormalization, but we have not calculated these graphs completely in arbitrary
dimension d. For computation the diagrams Σ5 and Σ6 with the required accuracy
we used the recurrence relations.
There are other techniques for calculation of such diagrams: usage of hypergeometric function expansions [1], the evaluation of complicated Feynman integrals
[2], the uniqueness method [3], reduction to master integrals [4,5] etc.
References
1. Tobias Huber, Daniel Maetre // Computer Physics Communications 178 (2008)
755-776.
2. Kotikov A.V. The Gegenbauer Polynomial Technique: the evaluation of complicated Feynman integrals // arXiv:hep-ph/0102177v1 14 Feb 2001.
3. Kazakov D.I. Many-loop calculations: the uniqueness method and functional equations, Joint Institute for Nuclear Research, Dubna. Translated from
Teoreticheskaya i Matematicheskaya Fizika, Vol.62, No.l, pp.127-135, January,
1985. Original article submitted January 16, 1984.
4. Baikova P.A., Chetyrkin K.G. Four Loop Massless Propagators: an Algebraic
Evaluation of All Master Integrals // arXiv:1004.1153v2 [hep-ph] 20 Apr 2010.
5. Chetyrkin K.G., Tkachov F.V. // Nuclear Physics B192 (1981) 159-204.
47
Canonical formalism for embedding theory with partial
gauge fixing
Semenova Elizaveta
derenovacio@mail.ru
Scientific supervisor: Dr. Paston S.A., Department of High
Energy and Elementary Particle Physics, Saint-Petersburg State
University
The research is focused on the problems related with the description of gravitation. The formulation of the gravity theory first suggested by Regge and Teitelboim
where the space-time is a four-dimensional surface in a flat ten-dimensional space
is considered. Canonical formalism for embedding theory with partial gauge fixing that match time of the surface and time of the space is developed. The results
obtained will be used for developing the canonical formalism for formulation of
gravity as a field theory in the ambient space. In the future, it may help to avoid
some of the problems arising in attempts to quantize the gravity.
Introduction
The Einstein's General Relativity is a common theory of gravity. It works well
for classical physics, but all attempts of quantum gravity theory construction have
failed. In 1975 T. Regge and C. Teitelboim proposed an alternative way of gravity
description - The Embedding Theory [1]. According to this theory the independent
variable is the embedding function. This function defines four-dimensional surface
in flat ten-dimensional space
y a ( x µ ) : R 4 → R1,9 (1)
It allows us to avoid some problems connected with curvature of space. In
Regge and Teitelboim's approach, the standard Einstein-Hilbert's action
S = ∫d 3 x − g R (2)
is taken as an action of the theory. Induced metric expressed through the embedding function
g µν = ∂ µ y a ∂ ν ya (3)
is substituted into this action.
Equations of motion of this theory
a
G µν bµν
= 0
(4)
are called the equations of Regge and Teitelboim (R-T).
These equations are more general than the Einstein's one. In order to eliminate
extra decisions of R-T equations imposition of additional Einstein's constrains
Gµ ⊥ = 0 (5)
were suggested.
The embedding theory is not free from problems connected with quantization of gravity. Foliation theory [2] could be the next step on the way of solving
such problems. This theory deals with many 4-dimentional surfaces each of them
describes dynamics of 3-dimensional space in the flat ambient space. These 4-di48
mentional surfaces pass through each point of R1,N-1 space, they neither intersect nor
interact. We can say that R1,N-1 is foliated on a system of surfaces. All disturbances
propagate along the surfaces. Geometry of surfaces corresponds to the solution of
R-T equations. This theory formulates gravity as a field theory in the flat ambient
space. Quantization of such type theory is well known. To construct the canonical
formalism for foliation theory we need to consider conventional embedding theory
with additional condition (gauge fixing)
y0 = x0 (6)
This consideration is the object of this work.
Canonical formulation
The addition of gauge condition into the action can lead to loss of some motion
equations. In this case we have only nine R-T motion equations (instead of ten)
G µυ bµνA = 0. (7)
It was verified that the tenth equation
(0)
G µν bµν
=0
(8)
automatically satisfies due to satisfying of nine equations and presence of coordinate
condition. Therefore, our gauge condition does not spoil the theory.
The action under satisfied gauge condition takes the following form, with
explicitly defined dependence on the time derivatives:

 3
y A BAB y B
1
S = ∫dx 
+ 1 + y A Π ⊥ AB y B BDD  ,
(9)
3

2
A
B
 1 + y Π ⊥ AB y

here
3 A
Π ⊥ B y B
A
n =
(10)
3
1 + y A Π ⊥ AB y B
The gauge fixing leads to reducing the number of constraints by one. Since this
theory is not invariant under time substitution, our Hamiltonian
1
1
H = ∫d 3 x π A y A − L = ∫d 3 x
nA B AD nD − BDD (11)
3
2
1 + y A Π ⊥ AB y B
where
δL
1
π E = E = BFE n F + nE BDD − n F BFH n H (12)
2
δy
(
(
)
(
)
)
is not equal to zero. We cannot obtain explicit expression for y(p) in terms of yA
and πA, hence we cannot directly realize the canonical formalism.
However, this can be done by imposing the above-mentioned Einstein's constraints. In terms of the variables of the theory and taking into account the gauge
fixing, they take the form:
1
0 = nA B AD nD − BDD (13)
2
A
3
3
3


1
i =
− g D k  Lik , lm blm nA  (14)



(
)
49
Hamiltonian appeared to be proportional to one of these constraints. So we consider the embedding theory with imposing Einstein's constrain and partial gauge
fixing.
Since
n A = B −1 AB π B (15)
all of constraints can be expressed as a function of coordinates yA and momenta πA:
3
Φ i = π A eiA (16)
1
0 = π A π B B −1BA − BDD (17)
2


3 3
1 3 −1ik 
bA πA . i = − − g D k 
(18)
3


 −g

(
)
Constraint algebra
It is convenient to deal with convolution of constraints and arbitrary functions.
It simplifies expressions and leads to more laconic form of results:
Φ ξ ≡ ∫d 3 x Φ i (x )ξi (x ) (19)
ξ0 ≡ ∫d 3 x  0 (x )ξ (x ) (20)
3 −1ik
3
ξ ≡ ∫d 3 x i (x )ξi (x ) = ∫d 3 x b A π A D k ξi (x ) It is more convenient to consider the action of constraints on
3
πA
−g than πA, because (22) is a scalar under 3-dimensional transformations.
The Poisson brackets of constraint Φξ and canonical variables are:
Φ ξ , y A ( x) = ξi (x )∂i y A (x) 

π A (x ) 
π A (x )

i
Φ
.
 ξ,
 = ξ (x )∂i
3
3


−
−
g
x
g
x
(
)
(
)


{
}
(21)
(22)
(23)
(24)
So it was found that Φξ generates a transformation of 3-dimensional coordinates
x i→x i+ξ i(x).
If some tensor is constructed out of y A and (17), then we can get the Poisson
brackets of it and constraint Φξ:
Φ ξ , a = ξ i ∂ i a, (25)
Φ ξ , a k = ξi ∂i a k − a i ∂i ξ k
(26)
{ }
{ }
{Φ , a }= ξ ∂ a
ξ
k
i
i
k
+ ai ∂ k ξi
50
(27)
Hence we can get rules for some tensor density. Since all of constraints are
tensor densities we can write following Poisson brackets:
3
 i 3 k
i
k
3
Φ
Φ
=
−
Φ
ξ
,
d
x
D
i ζ − ζ Di ξ 

k
ξ
ζ
∫
(28)


{
}
{Φ ,  }= −∫d x   ξ D ζ + ζ D
{Φ ,  }= −∫d x  ξ ∂ ζ ξ
3
ζ
k
i
0
ζ
3
3
3
i
k
0 i
i
k

ξi  
(29)
(30)
i
It was found out that constraints Φξ and Hξ have a similar effect on 3gik. So in
the future the combination of these constraints can be considered
Ψ ξ = ∫d 3 xΨ k (x )ξ k (x ) (31)
where
3 ik
Ψk = k + Φi g (32)
The main advantage of this value is that {Ψξ, 3gik}=0. Further it is necessary to
calculate the Poisson brackets {Hξ, H𝜁}, {Hξ0, H𝜁} and {Hξ0, H𝜁0} then constraint
algebra will be constructed.
Conclusion
Canonical formalism for embedding theory with partial gauge fixing that
matches time of the surface and time of the space is developed. The obtained constraint algebra built using the gauge fixing, will be compared with one obtained
without it in work [3]. The results obtained will be used for developing the canonical
formalism for formulation of gravity as a field theory in the ambient space (foliation theory) [2]. In the future, it may help to avoid some of the problems arising
in attempts to quantize the gravity.
References
1. Regge T., Teitelboim C. General relativity a la string: a progress report. In
Proceedings of the First Marcel Grossmann Meeting, Trieste, Italy, 1975. Ed. R.
Ruffini, North Holland, Amsterdam, 1977. p. 77.
2. Paston S.A. // Theor. Math. Phys., v.169, N 2, pp. 1611-1619, 2011,
arXiv:1111.1104.
3. Paston S.A., Semenova A.S. // Int. J. Theor. Phys., v.49, N 11, pp. 2648-2658,
2010, arXiv:1003.0172.
ξ
51
H. Biophysics
UVC-induced DNA destructions in vitro
Parr Marina
mparr@mail.ru
Scientific supervisor: Dr. Paston S.V., Department of Molecular
Biophysics, Faculty of Physics, Saint-Petersburg State
University
Introduction
It is known that the direct influence of the UV-light results in modifications of
the DNA structure. There are six main photochemical reactions that take place in
DNA: photodimerization of thymine, photodimerization of cytosine, dimerization
of different nitrogenous bases, tautomerization of bases, intermolecular cross linking and the disruption of double helix [1]. The sensitivity of the DNA to UV-light
depends on the conditions of the experiment.
Results and Discussion
At the present work we studied the role of the ionic strength of the solution in
the UVC-induced changes in the DNA spectral properties. The experiment followed
this following scheme: using the sodium salt of the DNA, we have prepared the
DNA solutions of different ionic strengths (0,003М, 0,005М, 0,15М, 1М, 3М).
The concentration of DNA in all solutions was about 0,005%. These solutions
were exposed to UV-radiation. As the source of UV-light the low pressure mercury
lamp was used (λirr=254 nm). During the experiment we were measuring the time
of irradiation (t) that is unambiguously related with doze of irradiation. When the
solutions got some doze of irradiation, their spectra of absorbance were measured.
As the result we got the spectra of the DNA solutions with different ionic strengths
that were measured after they had got some doze of irradiation (Fig. 1).
0,005M
0,5
0,0
220
240
260
λ, nm
280
3M
1,0
D
D
1,0
300
0 min
60 min
120 min
180 min
240 min
300 min
0,5
0,0
220
240
260
λ, nm
280
300
Fig. 1. The spectra of the DNA solutions (minus the scattering i.e. the optical density of all systems at 310 nm) with the ionic strengths 0,005M and 3M after different time of irradiation.
It is convenient to trace the spectral changes at the wavelengths 230 and 260 nm.
To explain the results received it is necessary to compare the spectra of hydrolyzed
54
1,4
0,005M t=0min 1,4
1,2
1,2
1,0
1,0
0,8
0,8
D
D
and non-hydrolyzed DNA (Fig. 2). There is significant hyperchromic effect in case
of the hydrolysis of non-irradiated DNA, and, in contrast, slight hyperchromic effect for DNA irradiated with maximum doze of UV-light. It indicates a significant
disruption of the DNA secondary structure as a result of UV-irradiation.
0,6
0,6
0,4
0,4
0,2
0,2
0,0
220
240
260
280
300
0,0
220
0,005M t=300min
before hydrolisis
after hydrolisis
240
λ, nm
260
280
300
λ, nm
Fig. 2. The spectra of non-irradiated and irradiated with maximum doze DNA solution with ionic strength 0,005M before and after hydrolysis. These spectra are
reduced to the same DNA concentration. Hyperchromic effect is 42% for non-irradiated DNA and 6% for DNA irradiated with maximum dose.
When we destroyed the secondary structure of DNA by hydrolysis, we observed
the changes associated only with the dimerization of thymine: abrupt decreasing
of intensity at 260 nm and slight decreasing at 230 nm (Fig. 3).
0,4
0,005M
0,4
D
0,3
D
0,3
0,2
0 min
300 min
0,2
0,1
0,1
0,0
220
3M
240
260
λ, nm
280
300
0,0
220
240
260
280
300
λ, nm
Fig. 3. The spectra of hydrolyzed DNA solutions with ionic strengths 0,005M and
3M before and after irradiation with maximum doze of UV-light.
As the result of UV-irradiation of thymine in aqueous solution the decreasing
of its optical density at the maximum of absorbance can be seen [1]. From the data
given in [1] it is possible to estimate the ratio of the molar extinction coefficients
of thymine to the one of thymine dimer at different wavelengths:
 ε thym. 
 ε thym. 
= 60, 
= 1, 4
ε


 thym.dym.  λ = 260 nm
 ε thym.dym.  λ = 230 nm
The same changes were observed in the spectral properties of UVC-irradiated
oligonucleotide consisting of 16 units of thymidine monophosphate. It does not
55
have the secondary structure so all changes we see are the evidences of photodimerization of thymine [2].
It is known that the quantum yield of photodimerization of thymine is much
bigger than the quantum yields of other photoproducts [3]. So we can conclude that
in the absorption spectra of non-hydrolyzed DNA after UV-irradiation we observe
combined effect of two factors: dimerization of part of thymine bases, that leads
to a significant decrease of absorption at λ = 260 nm and causes no changes (or
causes slight decrease) of the absorbance at λ = 230 nm; disruption of the secondary
structure of DNA, that leads to the increase of the absorbance at all wave lengths
in the absorption spectrum. These conclusions explain the results obtained: there
is an increase of optical density at 230 nm (these changes are caused only by the
denaturation of DNA) and at 260 nm the changes of absorption are slight (the
reason of it is that at this wave length we observe the influence of two concurrent
factors - sharp drop in the intensity due to thymine dimerization and increase in
the intensity due to denaturation).
We have also measured the concentration of different nitrogenous bases in
the solutions of different ionic strengths as a function of the received doze of UV
by the Spirin's method, using the optical densities of hydrolyzed solutions at 270
and 290 nm [4]. Taking into account that dimers of thymine do not absorb light at
λ>240nm, we can consider that they will not give any impact in calculated concentration. So we will estimate the concentration of all nitrogenous bases except
thymine dimers and the molar extinction coefficients of UV-irradiated DNA at
260 nm (Table 1).
The results demonstrate that the concentration of nitrogenous bases in all studied
solutions decreases with the increase of the dose of UV. This effect is almost independent on the ionic strength of the solution. The absorption of the UV-irradiated
DNA at 260 nm is also determined by the amount of all nitrogenous bases except
the dimmers of thymine. Using the values of the concentration of nitrogenous bases,
we have calculated the molar extinction coefficients of UV-irradiated DNA in the
solutions of different ionic strengths depending on the dose of UV. The values of
molar extinction coefficient at long times of exposure indicate the destruction of
the secondary structure of DNA (Table 1). This effect dependents insignificantly
on the ionic strength of the solution.
56
Table 1. Relative changes of the concentration of nitrogenous bases (C0-C)/C0 (C0
– the concentration of nitrogenous bases in non-irradiated solution) and molar
extinction coefficients ε260(P) (l/(mol*sm)) at different time of irradiation (t)).
0,003M
0,15M
3M
t, min
(C0-C)/C0
ε260(P)
(C0-C)/C0
ε260(P)
(C0-C)/C0
ε260(P)
0
1
6300
1
6400
1
6200
30
0,99
6200
0,97
6400
0,96
6400
90
0,93
6600
0,92
6800
0,9
6700
180
0,86
7300
0,89
7000
0,85
7000
240
0,82
7700
0,82
7700
0,8
7800
300
0,71
8800
0,79
8000
0,76
7900
Conclusions
In the absorption spectra of non-hydrolyzed DNA after UV-irradiation we observe a combined effect of two factors: 1) dimerization of part of thymine bases,
that leads to a significant decrease of absorption at λ = 260 nm and causes no
changes (or causes slight decrease) of the absorbance at λ = 230 nm; 2) disruption
of the secondary structure of DNA, that leads to the increase of the absorbance
at all wave lengths in the absorption spectrum. These effects are quite slightly
dependent on ionic strength.
The work was supported by Russian Foundation for Basic Research (RFBR,
grants 12-08-01134, 13-03-01192) and the Ministry of Education and Science of
Russian Federation.
References
1. Smith K., Hanawalt Ph. // Molecular Photobiology. Academic Press, NY,
London (1969).
2. Mu W., Zhang D., Xu L., Luo Zh., Wang Y. // J. Biochem. Biophys. Methods,
v. 63, p. 111 (2005).
3. Rubin A.B. // Biophysics, v.2. Moscow, Knizhnyi Dom “Universitet” (2000).
4. Spirin A.S. // Biokchimiya (USSR), v. 23, p. 656 (1958).
57
Table of Content
A. Chemistry....................................................................................................... 5
Development of the method for determining the reaction products of organic
compounds occurring in laser-induced deposition of copper from solution
Zhigley Elvira, Safonov Sergey............................................................................ 6
D. Solid State Physics......................................................................................... 7
Characterization of pristine and fluorinated nanodiamonds by X-ray absorption
spectroscopy
Zagrebina Elena................................................................................................... 8
E. Applied Physics............................................................................................ 15
Migration of the radionuclides from nuclear accidents in the forest ecosystem
Merzlaya Anastasia............................................................................................ 16
G. Theoretical, Mathematical and Computational Physics......................... 21
Long-range rapidity correlations in a two-component model
Andronov Evgeny............................................................................................... 22
Renormalization group analysis of the inertial-range behaviour of a passive vector
field in a random shear flow
Gulitskiy Nikolay................................................................................................ 26
Splint in case of special embeddings
Kakin Polina...................................................................................................... 31
Critical behavior of the O(n)-φ4 model with an antisymmetric tensor order
parameter
Lebedev Nikita................................................................................................... 34
Spectral properties of quasi-periodic Schrodinger equations
Martemyanov Andrey......................................................................................... 39
Calculation of self-energy diagrams in the φ4 theory with help of recurrence
relations
Pismenskii Artem............................................................................................... 43
Canonical formalism for embedding theory with partial gauge fixing
Semenova Elizaveta........................................................................................... 48
58
H. Biophysics.................................................................................................... 53
UVC-induced DNA destructions in vitro
Parr Marina....................................................................................................... 54
Table of Content .............................................................................................. 58
59