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Slepton production in a background electromagnetic field
Volume 249, number 3,4
PHYSICS LETTERS B
25 October 1990
Slepton production in a background electromagnetic field A.V. Kurilin Department of Physics, MGZPI, VerhnyayaRadishevskaya 18, SU-I09 004 Moscow, USSR Received 13 July 1990
Scalar-lepton production via the decay ~ induced by a constant homogeneous electromagnetic field is considered. Asymptotic estimations of the decay-width in different domains are analyzed.
Supersymmetric theories [ 1 ] proposing a symmetry of bosons and fermions are believed to be the most promising candidates for new physics beyond the standard model. This belief is based on purely theoretical motivations, because there are no phenomenological indications that supersymmetry (SUSY) might be relevant for a description of nature. One reason why SUSY has fascinated many physicists is that it gives rise to a special ultraviolet behaviour of supersymmetric field theories and provides a neat solution of the hierarchy problem [2]. Besides, in a local version of SUSY (supergravity) supersymmetric transformations are related to the space-time transformations of general relativity, thus enabling one to construct an acceptable quantum theory of gravity [ 3 ]. All this makes SUSY a very attractive idea and compels one to find experimental evidence for it. Supersymmetric theories predict that all particles must appear in pairs of identical quantum numbers differing in spin by one half. For example, in a minimal supersymmetric version of the standard model [4 ] there are two spin-zero particles (sleptons) denoted usually by ~'R and ~L for each generation ofleptons ~. Apart from that, all gauge vector bosons are accompanied by neutral massive Majorana fermions such as the superpartner of the photon, the photino ?. In the limit of exact SUSY all ordinary particles and their partners have the same masses. However, in nature, SUSY, if it exists, is broken and the superpartners must be heavy enough to have avoided detection so far. Experimental observation of these will be one of the simplest ways to discover SUSY. Dif-
ferent suggestions have been given in the literature for detecting charged sleptons in e+e -, pp, e - p machines [5]. A considerable amount of work has already been carried out in both the experimental and theoretical directions but the situation is still unclear. The purpose of this paper is to study the effect of slepton production via the decay ~-~'~ induced by a constant homogeneous electromagnetic field. The physics of quantum processes in external electromagnetic fields is of great interest because it has many applications in the investigations of charged particles' interactions with monocrystals [6], of astrophysics [ 7 ], cosmology [8 ], etc. The most powerful theoretical tool in this research is the Furry perturbative approximation making use of exact solutions of relativistic particles' wave equations in the presence of a background electromagnetic field Fu~. Within this method the total width of the decay ~ ~'~ can be obtained by employing the well known Volkov wave functions [9]. The result of straightforward calculations can be written in the form
F( ~--*'~ ) = ½am~ i
× j d u { n [ A i ( z ) G i ' (z) - A i ' ( z ) G i ( z ) ] f (u) o
+Ai(z)f2 (u) + A i ' (z)f3(u) },
(1)
with the abbreviations
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
455
Volume 249, number 3,4
fl(U)=½(1--E~,)U+ A(u)
PHYSICS LETTERS B
½( l + e ~ I I ) ( 2 2 - - 2 1 + l - - u ) ,
=¢~ [zu~( 1 - u ) ] "3 ,
f3(u) = - (1 +eft,) [Zeu( 1 - u ) 2] 1/3
(2)
Special (Airy type) mathematical functions entering into eq. (1) have the following integral representations [ 10 ]: Ai(z)= 1 i dtcos(tz+}t rc
3)
0
Gi(z)=~
'
dtsin(tz+~t3),
dAi(z) Ai'(z)= dz ' Gi'(z)=
dGi(z) dz '
0
(3) with the argument
z= [Zu2(1-u)] -2/3 .
(4)
Our general expression (1) corresponds to a semiclassical approximation of the total width in the domain of comparatively weak external fields, when one can neglect corrections being proportional to the small parameters [g~ I << 1, [g21 << 1 gl = ¼e2m~aFu, FI~u , g2 = - ~eZm~4Fuu ffuu ,
where F ' " = ½eu"~°F~o. In such a case there is F y d e pendence of F ( ~ ) only through the parameter ~ 2,
(6)
which defines the background field strength in the rest frame of the initial lepton ~ (p~= m ~ ) 2. 2 Polarization effects of the decay are described by the lepton's parallel l~l i and orthogonal l~z mean helicities related with the polarization four-vector su in the following way:
=-1,
for ~--~L~.
ffU~P,) •
(8)
The masses of superparticles entering into (2), (4) through ratios 2 1 ~-
(M¢/m~) 2 ,
2 2=
(9)
(m?/m~) 2 ,
are model conditioned. Depending upon whether global or local SUSY is considered there exist two distinct types of mass phenomenology. Experimental information available at present allows only to give the lower limits M ¢ > 2 2 GeV, M?>0.5 GeV [12]. On this account it is quite permissible to treat 2 j, 22 as free parameters assuming in addition 21 >> 1. The general formula for total width ( 1 ) is rather complicated because the integration over the spectral variable - (V~wq ~) 2
(lO)
(with q" being the slepton's m o m e n t u m ) cannot be done analytically. For the sake of simplicity let us consider asymptotic estimations of the total width ( 1 ) in different domains. Supposing the slepton to be much heavier than the photino (21 >>22) one gets /'~--~ l a m ~ 2 1
(1-4-6¢1t )K l / X / 3 exp( --x/~/xl ) ,
F = lotrn~21 ( 1 + e~,) -~F( -~) (3x,)2/3, forxl >>1 ,
(11)
with x~=X2F 3/2, F(~)=1.354... ( F is the Euler gamma function). In the opposite case of the photino being the heaviest particle (21 < < 2 2 ) m o r e tedious calculations lead to F=½otm~22(l+e~ll)K
2
exp
-
K2X] 22 J '
for K2 << (2 1 / 2 2 ) 1/2 , F = ½am~22( 1 ...1_E~LI 1 2
~11= e Z z - Z m ~ S ( s u F I ~ F , ~ . P "~) ,
(7)
The multiplier E in eq. (2) stands for + 1 regarding what mode of the decay is considered: ~l For the electron it is the well known invariant of the quantum synchrotron radiation theory [ 11 ]. 456
for ~ o ~ a ? ,
for tel << 1 , (5)
~l = - e z - l m F 3 ( s ,
e=+l,
u= 1-ez-'m~34
X [21u+22(1-u)-u(1-u)]
z=em~3xS-(Fu,p")
25 October 1990
for (21/22) 1/2 << K2 << 1 , F = ½am~22( 1 + e~LI)~TF( } ) (3x2)2/3, for xz >> 1 ,
(12)
with x2 =)~22 -3/2 • The intermediate region of the slepton and the
Volume 249, number 3,4
PHYSICS LETTERS B
25 October 1990
~? in the d o m a i n o f relatively weak fields (Fu,,<< m2/e). T h i s is d u e to the factor o f e x p o n e n t i a l s u p p r e s s i o n which is typical for processes i n d u c e d by an external field [ 13]. U n f o r t u n a t e l y the region o f greatest physical interest ( Z ~ 10 J4) is n o t accessible n o w by e x p e r i m e n t . T h e results o f n u m e r i c a l c o m p u t a t i o n s in the case o f the electron are s h o w n in fig. 1. However, as a final r e m a r k let us stress that in the v i c i n i t y o f astrophysical objects with extremely high e l e c t r o m a g n e t i c fields this process m i g h t very well be observed. F o r instance, in theoretical papers d e v o t e d to pulsars [ 14 ] it is suggested that m a g n e t i c fields o f sufficient o r d e r in the characteristic q u a n t u m elect r o d y n a m i c value m2/e~ 1013G p r o b a b l y exist in nature. F u r t h e r details o f astrophysical a p p l i c a t i o n s will be c o n s i d e r e d in a f o r t h c o m i n g p u b l i c a t i o n .
10 t°
10-'c
10-2[
References
I0 -31
10 is
1014
10 is
1016
%
Fig. 1. Illustrative estimations showing how the time of decay depends upon the parameter Z. The curves represent the following choices: ( 1 ) Me = 50 GeV, m, = 0 GeV. (2) Me = 100 GeV, rn,= 100 GeV, (3) Me=50 GeV, rn~= 500 GeV. p h o t i n o h a v i n g c o m p a r a b l e masses can be characterized by the expression F = 2~cern~2( 1 +E~l I)1c/x//2 exp( -- 9/21c) , for x<< 1 , F = ~70~m~2 ( 1 + e~,) ½F( 32) ( 3 x ) 2/3 , for x>> 1 ,
(13)
which is valid p r o v i d e d that 12~ - 2 2 1 <<2--- ~ (2j +
•2)" F r o m a n i n s p e c t i o n o f f o r m u l a s ( 11 ) - ( 1 3 ) one c o n c l u d e s that regardless o f the mass ratio rn,/M¢ it is very hard to p r o d u c e a s l e p t o n b y the r e a c t i o n 2--+
[ 1] See, e.g., H.P. Nilles, Phys. Rep. 110 (1984) 1. [2] E. Witten, Phys. Lett. B 105 ( 1981 ) 267. [3 ] See, e.g., P.C. West, Introduction to supersymmetry and supergravity (World Scientific, Singapore, 1990). [4] See, e.g., R.N. Mohapatra, Unification and supersymmetry (Springer, Berlin, 1986 ). [ 5 ] See, e.g., H.E. Haber and G.L. Kane, Phys. Rep. 117 ( 1985 ) 75. [6] V.N. Baier, V.M. Katkov and V.M. Stakhovenko, Usp. Fiz. Nauk 159 (1989) 455. [ 7 ] A.A. Sokolov and I.M. Ternov, Relativistic electron (Nauka, Moscow, 1983). [8 ] A.D. Linde, Elementary particle physics and inflationary cosmology (Nauka, Moscow, 1990 ). [9] D.M. Volkov, Z. Phys. B 94 (1935) 250. [10]M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Applied Mathematical Series, 55, 1964 ) (National Bureau of Standards, Washington). [IIIA.A. Sokolov and I.M. Ternov, Synchrotron radiation (Akademie Verlag, Berlin, 1968). [ 12 ] R. Barbieri, I. Ferrante, F. Fidecaro, M. Frigeni and J.-F. Grivaz, CERN scientific report 8 (1989) 121. [ 13 ] See, e.g., A.A. Sokolov, I.M. Ternov, A.V. Borisov and V.Ch. Zhukovskii, Phys. Lett. A 49 (1974) 9. [ 14] See, e.g., S.L. Shapiro and S.A. Teukolsky, Black holes, white dwarfs, and neutron stars (Wiley, New York, 1983 ).
457
PHYSICS LETTERS B
25 October 1990
Slepton production in a background electromagnetic field A.V. Kurilin Department of Physics, MGZPI, VerhnyayaRadishevskaya 18, SU-I09 004 Moscow, USSR Received 13 July 1990
Scalar-lepton production via the decay ~ induced by a constant homogeneous electromagnetic field is considered. Asymptotic estimations of the decay-width in different domains are analyzed.
Supersymmetric theories [ 1 ] proposing a symmetry of bosons and fermions are believed to be the most promising candidates for new physics beyond the standard model. This belief is based on purely theoretical motivations, because there are no phenomenological indications that supersymmetry (SUSY) might be relevant for a description of nature. One reason why SUSY has fascinated many physicists is that it gives rise to a special ultraviolet behaviour of supersymmetric field theories and provides a neat solution of the hierarchy problem [2]. Besides, in a local version of SUSY (supergravity) supersymmetric transformations are related to the space-time transformations of general relativity, thus enabling one to construct an acceptable quantum theory of gravity [ 3 ]. All this makes SUSY a very attractive idea and compels one to find experimental evidence for it. Supersymmetric theories predict that all particles must appear in pairs of identical quantum numbers differing in spin by one half. For example, in a minimal supersymmetric version of the standard model [4 ] there are two spin-zero particles (sleptons) denoted usually by ~'R and ~L for each generation ofleptons ~. Apart from that, all gauge vector bosons are accompanied by neutral massive Majorana fermions such as the superpartner of the photon, the photino ?. In the limit of exact SUSY all ordinary particles and their partners have the same masses. However, in nature, SUSY, if it exists, is broken and the superpartners must be heavy enough to have avoided detection so far. Experimental observation of these will be one of the simplest ways to discover SUSY. Dif-
ferent suggestions have been given in the literature for detecting charged sleptons in e+e -, pp, e - p machines [5]. A considerable amount of work has already been carried out in both the experimental and theoretical directions but the situation is still unclear. The purpose of this paper is to study the effect of slepton production via the decay ~-~'~ induced by a constant homogeneous electromagnetic field. The physics of quantum processes in external electromagnetic fields is of great interest because it has many applications in the investigations of charged particles' interactions with monocrystals [6], of astrophysics [ 7 ], cosmology [8 ], etc. The most powerful theoretical tool in this research is the Furry perturbative approximation making use of exact solutions of relativistic particles' wave equations in the presence of a background electromagnetic field Fu~. Within this method the total width of the decay ~ ~'~ can be obtained by employing the well known Volkov wave functions [9]. The result of straightforward calculations can be written in the form
F( ~--*'~ ) = ½am~ i
× j d u { n [ A i ( z ) G i ' (z) - A i ' ( z ) G i ( z ) ] f (u) o
+Ai(z)f2 (u) + A i ' (z)f3(u) },
(1)
with the abbreviations
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
455
Volume 249, number 3,4
fl(U)=½(1--E~,)U+ A(u)
PHYSICS LETTERS B
½( l + e ~ I I ) ( 2 2 - - 2 1 + l - - u ) ,
=¢~ [zu~( 1 - u ) ] "3 ,
f3(u) = - (1 +eft,) [Zeu( 1 - u ) 2] 1/3
(2)
Special (Airy type) mathematical functions entering into eq. (1) have the following integral representations [ 10 ]: Ai(z)= 1 i dtcos(tz+}t rc
3)
0
Gi(z)=~
'
dtsin(tz+~t3),
dAi(z) Ai'(z)= dz ' Gi'(z)=
dGi(z) dz '
0
(3) with the argument
z= [Zu2(1-u)] -2/3 .
(4)
Our general expression (1) corresponds to a semiclassical approximation of the total width in the domain of comparatively weak external fields, when one can neglect corrections being proportional to the small parameters [g~ I << 1, [g21 << 1 gl = ¼e2m~aFu, FI~u , g2 = - ~eZm~4Fuu ffuu ,
where F ' " = ½eu"~°F~o. In such a case there is F y d e pendence of F ( ~ ) only through the parameter ~ 2,
(6)
which defines the background field strength in the rest frame of the initial lepton ~ (p~= m ~ ) 2. 2 Polarization effects of the decay are described by the lepton's parallel l~l i and orthogonal l~z mean helicities related with the polarization four-vector su in the following way:
=-1,
for ~--~L~.
ffU~P,) •
(8)
The masses of superparticles entering into (2), (4) through ratios 2 1 ~-
(M¢/m~) 2 ,
2 2=
(9)
(m?/m~) 2 ,
are model conditioned. Depending upon whether global or local SUSY is considered there exist two distinct types of mass phenomenology. Experimental information available at present allows only to give the lower limits M ¢ > 2 2 GeV, M?>0.5 GeV [12]. On this account it is quite permissible to treat 2 j, 22 as free parameters assuming in addition 21 >> 1. The general formula for total width ( 1 ) is rather complicated because the integration over the spectral variable - (V~wq ~) 2
(lO)
(with q" being the slepton's m o m e n t u m ) cannot be done analytically. For the sake of simplicity let us consider asymptotic estimations of the total width ( 1 ) in different domains. Supposing the slepton to be much heavier than the photino (21 >>22) one gets /'~--~ l a m ~ 2 1
(1-4-6¢1t )K l / X / 3 exp( --x/~/xl ) ,
F = lotrn~21 ( 1 + e~,) -~F( -~) (3x,)2/3, forxl >>1 ,
(11)
with x~=X2F 3/2, F(~)=1.354... ( F is the Euler gamma function). In the opposite case of the photino being the heaviest particle (21 < < 2 2 ) m o r e tedious calculations lead to F=½otm~22(l+e~ll)K
2
exp
-
K2X] 22 J '
for K2 << (2 1 / 2 2 ) 1/2 , F = ½am~22( 1 ...1_E~LI 1 2
~11= e Z z - Z m ~ S ( s u F I ~ F , ~ . P "~) ,
(7)
The multiplier E in eq. (2) stands for + 1 regarding what mode of the decay is considered: ~l For the electron it is the well known invariant of the quantum synchrotron radiation theory [ 11 ]. 456
for ~ o ~ a ? ,
for tel << 1 , (5)
~l = - e z - l m F 3 ( s ,
e=+l,
u= 1-ez-'m~34
X [21u+22(1-u)-u(1-u)]
z=em~3xS-(Fu,p")
25 October 1990
for (21/22) 1/2 << K2 << 1 , F = ½am~22( 1 + e~LI)~TF( } ) (3x2)2/3, for xz >> 1 ,
(12)
with x2 =)~22 -3/2 • The intermediate region of the slepton and the
Volume 249, number 3,4
PHYSICS LETTERS B
25 October 1990
~? in the d o m a i n o f relatively weak fields (Fu,,<< m2/e). T h i s is d u e to the factor o f e x p o n e n t i a l s u p p r e s s i o n which is typical for processes i n d u c e d by an external field [ 13]. U n f o r t u n a t e l y the region o f greatest physical interest ( Z ~ 10 J4) is n o t accessible n o w by e x p e r i m e n t . T h e results o f n u m e r i c a l c o m p u t a t i o n s in the case o f the electron are s h o w n in fig. 1. However, as a final r e m a r k let us stress that in the v i c i n i t y o f astrophysical objects with extremely high e l e c t r o m a g n e t i c fields this process m i g h t very well be observed. F o r instance, in theoretical papers d e v o t e d to pulsars [ 14 ] it is suggested that m a g n e t i c fields o f sufficient o r d e r in the characteristic q u a n t u m elect r o d y n a m i c value m2/e~ 1013G p r o b a b l y exist in nature. F u r t h e r details o f astrophysical a p p l i c a t i o n s will be c o n s i d e r e d in a f o r t h c o m i n g p u b l i c a t i o n .
10 t°
10-'c
10-2[
References
I0 -31
10 is
1014
10 is
1016
%
Fig. 1. Illustrative estimations showing how the time of decay depends upon the parameter Z. The curves represent the following choices: ( 1 ) Me = 50 GeV, m, = 0 GeV. (2) Me = 100 GeV, rn,= 100 GeV, (3) Me=50 GeV, rn~= 500 GeV. p h o t i n o h a v i n g c o m p a r a b l e masses can be characterized by the expression F = 2~cern~2( 1 +E~l I)1c/x//2 exp( -- 9/21c) , for x<< 1 , F = ~70~m~2 ( 1 + e~,) ½F( 32) ( 3 x ) 2/3 , for x>> 1 ,
(13)
which is valid p r o v i d e d that 12~ - 2 2 1 <<2--- ~ (2j +
•2)" F r o m a n i n s p e c t i o n o f f o r m u l a s ( 11 ) - ( 1 3 ) one c o n c l u d e s that regardless o f the mass ratio rn,/M¢ it is very hard to p r o d u c e a s l e p t o n b y the r e a c t i o n 2--+
[ 1] See, e.g., H.P. Nilles, Phys. Rep. 110 (1984) 1. [2] E. Witten, Phys. Lett. B 105 ( 1981 ) 267. [3 ] See, e.g., P.C. West, Introduction to supersymmetry and supergravity (World Scientific, Singapore, 1990). [4] See, e.g., R.N. Mohapatra, Unification and supersymmetry (Springer, Berlin, 1986 ). [ 5 ] See, e.g., H.E. Haber and G.L. Kane, Phys. Rep. 117 ( 1985 ) 75. [6] V.N. Baier, V.M. Katkov and V.M. Stakhovenko, Usp. Fiz. Nauk 159 (1989) 455. [ 7 ] A.A. Sokolov and I.M. Ternov, Relativistic electron (Nauka, Moscow, 1983). [8 ] A.D. Linde, Elementary particle physics and inflationary cosmology (Nauka, Moscow, 1990 ). [9] D.M. Volkov, Z. Phys. B 94 (1935) 250. [10]M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Applied Mathematical Series, 55, 1964 ) (National Bureau of Standards, Washington). [IIIA.A. Sokolov and I.M. Ternov, Synchrotron radiation (Akademie Verlag, Berlin, 1968). [ 12 ] R. Barbieri, I. Ferrante, F. Fidecaro, M. Frigeni and J.-F. Grivaz, CERN scientific report 8 (1989) 121. [ 13 ] See, e.g., A.A. Sokolov, I.M. Ternov, A.V. Borisov and V.Ch. Zhukovskii, Phys. Lett. A 49 (1974) 9. [ 14] See, e.g., S.L. Shapiro and S.A. Teukolsky, Black holes, white dwarfs, and neutron stars (Wiley, New York, 1983 ).
457