SCIENTIFIC ABSTRACT SHAKHBAZYAN, V. A. - SHAKHGILDYAN, V. V.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001548530002-3
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: August 9, 2001
• Sequence Number: 2
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

83200
S/056/60/039:/002/037/044
B006/BO7O
AUTHOR: Shakhbazyan, V. A.
p
L A
TITLE. =nfr=are_~d Ca`tastr~ope In Scalar Quantum El'ectrodynam'
3-cs
PERIODICAL; Zhurnal eksperimentallnoy i teoreticheskoy fiziki, 1960,
Vol-. 39, No. 2(8), pp. 484 - 490
TEXT. While a method for the study of the asymptotic.behavior in the;
infrared in spinor electrodynamics-has been treated many times beforei,
there tcours an additional difficulty in the electrodynami6s of spin-zero
particles which is connected with the impossibility of taking into account
the four-boson interaction.. The author has shown in two previous papers
(Refs. 5,6) that the group of the multiplicative renormalizations in,
scalar electrodynamics have two charge invariants. In:the infraredi.the
photon propagation function is regular, and the charg e-- invariant, which
describes the electromagnetic interaction, is a const :ant (Pee Ref. 2).
The situation is essentially more complicated for the four-boson inter-
action. L. P. Gor'kov (Ref. 7) has shown already that.the,Green function
of the scalar meson has an infrared singularity.. The'author,of the present
Card 1/3

83200
infrared Catastrophe in Scalar Quantum Electro- S/056/60/039/002/037/044
dynamics BO06/hO
paper now investigates the asymptotic behavior of the four-vertexfunction'
of the spin-zero particle,in the infrared. The method Is based on the:
Feynman graph technique. Out of the graphs with four meson ends, only,
those are concerned in the investigation of the infrar'ed si ngulaIrity~of
the matrix structure considered which are shown in Fig. I is' first
shown that the scalar four-meson function has a l9garithmic
2
singularity when the squares of the external momenta tend toom it is
significant that depends on h and, therefore, the~behavl'or of the:
2
second.charge invariant hd 0 in the infrared has to~be taken into.
M 1
account. (d Greens function for the scalar meson) ;,'Later the author
deri-7es the functional and differential equations for~the determination
of the asymptotic behavior of the function 0 in inf, rared :and discusses
the possibilities of solving them by perturbation theoretical methods., In
the last part of the work the author discusses a procedure.of removIng
the infrared divergences and summation of the probabilities~for charged
meson-meson scattering involVing the emission of an' arbitrary numbers of
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83200
Infrar9d Catastrophe in Scalar Quantum Electro- S/056'60/039/002/037/044
dynamics Boo67BO70
soft quanta. In order to eliminate the infrared catastrophe in,the lows
orders of perturbation theory, the author;uses.the method of~.generalizing
the probability graphs, suggested by A. L. Abrikosov,. The thiee sets of
graphs which are relevant to the case considered here are represented~in
Figs. 2-4. The author thanks D. V. Shirkov for.guidance, and:,I.. F.Ginzb~urg,
L. P. Gor1kov, and L.-D. Solov'yev for discussions. There are 4 figures
and 8 Soviet references.
A SOCIATION: Matematicheskiy institut Akademii nauk SSSR (Mathematics
Institute of the Academy of Sciences.of the US~LRJ
SUBMITTEDi March 24, 1960
Card 3/3

i7
Q87
S/022/61/014/002/005/ooa:
B1 2 5/~205
9 S-1
D 43
AUTHOR: Shakhbazyan, V. A.
TITLE: Radiative corrections of loviest orders in scalar quantum
electrodynamics
PERIODICAL: Izvestiya Akademii nauk Armyanskoy SSR.., Seriya fiziko-.
matematicheskikh nauk, v. 14, no. 2, 1961, 79-69
TEXT: The present paper contains formulas of radiative corrections of
loviest orders to the Green functions of the meson and the photon, which
are valid for the entire domain of the argument (here,:,.the momentum),'and
presents a calculation of the doubly-logarit4ic ultraviolet asymptotic
behavior of the vertex part of the order of e and of the infrared
asymptotic behavior of the four-vertex function _11. The whole investi-
gation is performed in the Duffin-Kemmer formalism. Using them methods.-
'elaborated by N. N. Bogolyubov and D. V. Shirkov (Vveae,:niye teoriyu:
kvantovannykh poley. GITTL. M. 1957 (introduction into thejheory of:
quantized fields)) one obtains the following relation in second perturbation-
theoretical approximation:
-Card 1/17

-X~ lRd"
I I ~ I [: ;!I'll I I --1Tp1--T77!-7--,,P-rT.,- =vj
:11

21987
S/02 2/61/014/002/005/008,
Radiative corrections of ... B125/~205
In I p2 21>
712
A (p)~
P2 2,
17t
stands for the fictious.mass of the photon introduced for the purpose
0
of eliminating the infrared divergence. The expression':.for.7A(p) is
accurate up to the imaginary part. By determining the~arbitrary constants
from the conditions for the vanishing of the radiative corrections to
the outer linesp one obtains a definite expression With the aid
of the total Green function of the meson: G (p) (p)+ !~c (p) (p)G (p)
(1.2) one obtains
Card 3/17

I . : i : : I I . , 1 11 11 1 . I
I I I
1 11! 11 1.- 1 - I I' I:- , . 1i i:,
;
. z :!"
-Jill

M87
S/02 ~/61/014/002/005/008
Radiative corrections of ... B125/B205~
ei 2+M2 2
In P2
2 - 8 IT
ma m
2+ 2 2 M2 2 2
P.
and m P P + m ne (1.6),
_2P
4 In (d't-L i) +
~:P2 P_ _: I , ,
M4 P 2,_M2 M2
+ 8 In + + 8 (alot.- 1) +
m
P
~d 4 (3 - d(l)
0 (nZ2)
U2 2 2 2
P M- P + nz-
In.
1(2) (P)
p2
Eq. (1-3) is accurate up to third-order terms with res~pect:to e. For-
2 2
p m one obtains easily the expressions previously derived by the
author for a (p), b (p), d P), and
(2) (2) m
C
ard
6/17.,

21987
S/O~i/61/014/002/005/008~
Radiative corrections of ... B125/B205
holds with an accuracy up to the term -e Based on the same considera-
tions made in the well-known book by Bogolyubov and Shirkov,, one arrives
at the following parametric representation of the Green function of the
photon:
Cma k" k- 1 2 2IM2)/.
4m
D (k) gmn 4- d~W
(2)t, F k-+ iz 48rz M2 (/112-7 k2+ jZ)
2 2
2 e k 21 2
and d(k 1 + 1n + (1-13) holds for lk m The
2 a
48-,E m
second part of the present paper deals, with radiative.correct ions to.the
ations
vertex parts of and In calculating them, finite inteir
cannoVbe dispensed with. The author therefore confines himself to
calculating the doubly-logarithmic asymptotic-behavior!!
,:of thefunction, 1-2-1
2
in the lowest approximations with respect to e and h.,: Methods for the
determination of the doubly-lo-arithmic asymptotic behavior of the vertex
Card_8/17

?1987
S/02~/61/014/002/005/008~~
Radiative corrections of ... B125/~205
2
parl, of the order of e have been de-vised-by V. V. Sudakov (ZhETF, 30
1956, 87). The method worked out by the latter for the calculation
of the type inteGral is applied in quantum electrodynanics without any,
alteration; a difference will occur only in the investigation of the
matrix structures. The type integral then acquires the form
d4k'
(2.1), and for
2 2 2 +
1-(p-k' + i --(q-k, + i (k
2 2. 1 2! M2
!k ~pqf q -P (2.2) the asymptotic behavior.takes the
form J ln (2-3). The definite.exprdssion ford
jP_q 2
2
p
the vertex operator of second order reads
P k2
In in
.(2) (p, q. k) f
32-.,2. P
pn + qn'
X I n n f2.4)
__ =in X q~i
Card 9/17
1+11W ;,:TIFIFTIP~ M Cf

21987
S/022/61/014/002/005/008
Radiative corrections of ... B125/B205:
An infrared asymptotic behavior of the function exists if.the squares,
2
of the external momenta tend toward m It is evident,1hat the behavior
of the graphs with infrared singularities is essential.Jin this connection'.~
The radiation operators of fourth order for the scattering of particles
of equal sign have the form
64
<
(P') (q') p) 5,b
4 .4
(q. p). (p. q)
he2
lo)
2 YJ he. (Pit-IN +
(2.
W. q), (P" P)
J(2)
+ he, P1. P2) 6 (P' +q -p-q).~
W.
Card 10/ 17
14-~ ~3 Ut~'.

21987
S/022 61/014/002/005/0W
B205
B125/
Radiative corrections of
q)
84 (S(I) + S(")) elo p p
X
q)
(pl) a (q') P) ~4
X q, --p), (q'. p', q'- W
ip', qI, p'- q)
1" q, P). (Q' P.
P3)1 i(2.6)
J(2)
(pt. P
+
P'. q,, P. -p1. lp*. P, P, -.1)
3.aeci,
k.
p k) D" W d4
(p, (p., k) D" (p, k) D.. 0
All (PI, P..)
gel
k1 DI( k)
J(2) (p d4k, :(2.8)
(p, (pt - k)
(p, +-k)'
JM (PI, pt. P.,)~~ ~4m' 3(p
e.
Card 11/17

. I !( i i 1 1, OF I . .
.PF
]
IT

021,16 1/014/ow.'/oIj-,/0()0
~adiantive corrections of 31 ~5/3-205
Mcson iithin the Klein-Gordor. formalism, and Do stands for the Green
function of the photon. The usual calibration do = 1 is used. in
(2-5) and (2.6) summation is performed over the momentum grolIUPS given;
above', p,q are the initial momenta of the particles, and pl,ql are their
final momenta. The problem is to find the behavior of the integrals
2 2 12 t2 2
(2.10) for p 9 q 7 q I P M simultaneously in the neighbor-
hood of k - 0. Omitting all small terms in the numerators integrals of
4
r d k 2 2 2 2 2 2
the types P M k (k0) k
2 2 2
(2pi k-c, )(2P2k+a )k
4
d k 2 12 2 12 2
p q q P M. mu t be estimated
2 2 2 1 S
(2p k-a (2 k-oL )k
P2
in order to determine the infrared asymptotic behavior'of integrals
(2-7) - (2.10). By evaluating these integrals in the neighborhood of-~
(2)
k ---0 one obtains the following for
Card 13/17

i i f 11 , I I ; I I - I I I I ~ I I I f FT f-r , ; ~ , '..

~11.987
S/022/6!/014/002/005/006
Radiative corrections of ... B125/B205.11
A (Si S-. S: + s. In-I + 1f
-f 6-j V'S, 4s.-s, t +
1) (s, - I)IS1
S2 s2
--In (2.13~
4sa V1 s.. (I t s. 1) 1 j.S,1)
+ In
4 s, V, S., II -+I Z ~1) 1/-,Fs-., -A -1-+ -S3
(p 4- q) (p, -P)2 (p., q)2:
S1 --7 -4nj~ '13 (2.14):
4m"
S, + St + s,== I, S, > 1. 5.