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High-energy anomalous B-non-conservation - phantom or reality?
Volume 242, number 3,4
PHYSICSLETTERSB
14 June 1990
H I G H - E N E R G Y A N O M A L O U S B-NON-CONSERVATION - P H A N T O M OR REALITY? M.E. SHAPOSHNIKOV Institutefor Nuclear Research of the USSR Academy of Sciences, SU-117 312 Moscow, USSR
Received 28 March 1990
I discuss the recent suggestionthat processes with anomalous baryon number non-conservationcan be unsuppressed in high energy collisions. On the example of exactly solvable quantum mechanics I demonstrate that the implementation of the naive euclidean instanton technique gives a qualitatively wrongresult for the rate of the analogueof B-non-conservation.It is clarified how it is possible to have exponentially small cross sections in high energy collisions and an unsuppressed rate of B violation at high temperatures.
1. It is well known that the fermionic number is non-conserved in standard electroweak theory [ 1 ]. The reason is the anomaly in the fermionic current and the complicated vacuum structure in non-abelian gauge theories. Nevertheless, the baryonic and leptonic numbers are good labels for particle degrees of freedom under the usual conditions of low temperatures and densities. To have B-non-conservation, the system must tunnel through the potential barrier, separating vacua with different topological numbers. The height of the barrier coincides with the sphaleron mass Esph --~(3-5.4)Mw/otw [ 2 ]. So B-violation is suppressed by the tremendous semiclassical factor exp(-4n/O~w) [1 ]. This is definitively not the case if the temperature [3] or the density [4] of the system is high enough. Here the system can overcome the barrier classically provided the temperature (or the Fermi momentum for dense media) is compared with the effective height of the barrier. It is needless to say that this issue is important for early cosmology, in particular to the problem of the baryon asymmetry of the universe [3,5]. Moreover, one can expect strong B-non-conservation in decays ofsuperheavy technibaryons [6]. The natural intention is to address the question on the possibility of rapid B-non-conservation in high energy collisions [7 ]. The motivation is obvious: if the centre of mass energy exceeds the sphaleron mass one may naively expect that the exponential suppression characteristic for vacuum-vacuum transitions
will disappear. Interest to this problem was rapidly growing up after the calculation by Ringwald [ 8 ] (see also ref. [ 9 ] ) of the corresponding amplitudes in the instanton approximation. It has been found that this approximation definitely breaks down at high energies corresponding to the sphaleron mass. The main point of calculation takes multiple gauge and Higgs boson production into account, considerably enhancing the cross section of B-violation. The importance of the many-particle final state was previously discussed in ref. [10]. It was speculated in refs. [ 8,9,11,12 ] that the breakdown of the instanton approximation signals the invalidity of the small coupling expansion at high energies and indicates that the weak interactions become strong at X/~~Esph. This implies strong B-non-conservation, multiple W, Z and H production and other phenomena characteristic for strongly coupled theory [ 1 1,12 ]. The breaking of the instanton approximation at high energies is not specific for electroweak theory only. It was shown to take place in a number of twodimensional models of quantum field theory [ 1 1,13,14]. An important result was found in ref. [14] for the 0 ( 3 ) a-model with explicit symmetry breaking. It has been shown there that quantum corrections to the rate become large well before the centre of mass energy becomes equal to the sphaleron energy, namely at V/s~ gwEsph. So the cross section is exponentially small in the region of applicability of instanton calculus.
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
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The aim of this paper is to discuss this possibility in the framework of simple quantum mechanical (QM) models. I shall demonstrate that the implementation of the same technique, as was used in electroweak theory, also gives exponentially growth with energy rate of the nonperturbative transitions. However, the exact solution to the problem in QM models with several degrees of freedom keeps the exponential suppression for all initial energies. I conclude, therefore, that the breaking of instanton calculus at high energies signals the unsuccessful choice of a classical configuration to expand about, rather than strong coupling. In other words, strong anomalous B-nonconservation at high energies seems to be very unprobable. Nevertheless, rapid topological transitions at high temperatures take place. 2. Consider first the QM model of one particle in a double well potential. The euclidean action is
S=½~2+U(x),
(1)
U ( x ) = ~ ( x 2 - c 2 ) 2.
To explore the energy dependence of the amplitude we can address the question about the probability of transition from an exited state with n quanta in the left well to a state with the same number of quanta in the right well. Following refs. [8,9] we consider the euclidean functional integral for the multiparticle Green function:
G(tl .... , tzn) = f D[x]
exp(-S)(x(t,)...x(t2,)).
(2)
With the use of the instanton solution
xi.~t=ctanh(Mnt/2),
M2=22c 2 ,
(3)
we get in the lowest approximation
G=O f
dto ~
i=1
Sinst = 2C2MH ,
xi,st(ti-to)exp(-Si,st)
, (4)
where D stands for the gaussian functional integral around the instanton without zero modes, to is the position of the instanton. Zero mode normalization is included to D too. We can get the matrix element of transition Mn by (i) calculating the Fourier transform of (4) with exp(iZogit~), (ii) making an analyt494
14 June 1990
ical continuation to Minkowski time, to~i~oi, (iii) putting the Green function to the mass shell with the help of reduction formulae and taking into account statistical factors coming from the normalization of states. The "leg" amputation is given by ( 2 M n ) I / 2 x (e~- MH). The final result for the probability of transition is / " = O 2 ( 1 2 S i n s t ) 2n (n!) 2 e x p ( - 2Sinst) •
(5)
In the large n limit we get
F~-DZ2nexp{-2[Si,st-nlog(12eSi,~t/n)]). n
(6)
We see that in quantum mechanics, as in field theory, the instanton calculation gives exponentially growth with energy E (recall that n = E/Mn) "cross section" of anomalous fermion number non-conservation, hitting the "unitary" bound at E~SinstMH/ log( 12e) -- 1.5Esph, where = L,K sph' ~t,l~21wt21"4"2His the height of the barrier separating different vacua (sphaleron mass). It can be easily shown by calculating the corrections to eq. (2) that the instanton approximation breaks down at n -- (Sinst) !/2 i.e. at energies E~_ Esph/ (ref. [14] ). The explanation is obvious: the instanton configuration is a bad starting point to expand about at E~-E~ph. For a correct semiclassical treatment we should use a periodic euclidean configuration given by the equation dx
v/~[ U ( x ) _ E ] =t ,
(7)
explicitly involving the initial energy. This solution differs drastically from the initial instanton, even the boundary conditions are to be changed. Nevertheless, the result is qualitatively correct at energies ~ Esph. The reason is simple: for one degree of freedom all the energy is concentrated in a "right direction" coinciding with the negative mode of the sphaleron. Let us now go to a bit more complicated example with two degrees of freedom where the initial energy can be attributed to the "orthogonal direction". 3. Consider the QM model with two degrees of freedom x and y with the euclidean action
Volume 242, number 3,4
PHYSICS LETTERS B
With the use of the expression for normalizable zero fermionic modes
& = ½ ~ + iX(x ~ _ c ~) ~+ i p = + i X ( y ~ - c ~)
1
-{- ~ C 4 P [ ( X 2 - - C 2 ) 2 - ( y z - c 2 ) 2 1 2 .
(8)
It has four degenerate vacua at the classical level:
~o = N 1/2 [cosh (Mn t12 ) ] -4flMn ,
( 15 )
we get for the Green function describing the transition the following:
I++ >={x=c,y=c},
G = D ~ dto ~0 (t2n+ 1 - t o ) ~o(t2n+2 --to)
I + - > ={x=c, y = - c } ,
I-+ >={x=-c,y=c}, I--
14 June 1990
2n
) ={x=-c,y=-c}.
(9)
For p < 2 the sphaleron configurations are nothing but the points ( x = 0 , y = + c ) , ( y = 0 , x=_+c), E~ph= 1 Rat2 ~,* u~p 2 /~ 1 + p / 2 2 ) . Let us add fermions to the theory with the interaction
S f = f ( a * a - b'b) ( x + y) ,
( 1O)
× ~ Xinst(ti--to), i=1
(16)
in obvious notation. Using the same procedure as the one-dimensional Q M model we obtain the probability o f exclusive transition from the initial state i with fermion number non-conservation:
F=AfermD2 (6Sinst p ! ( n -X/2)2n p ) ! exp ( - 2Sinst)
(17)
where a, a*, b and b t are anticommuting operators, f is the Yukawa coupling. We shall have fermionic n u m b e r non-conservation together with a transition between the vacuum I + + ) and the vacuum I - - ). This transition is described by the instanton configuration
where Arerm is a factor coming from fermionic zero modes. It is not essential in what follows. Let us analyze this equation in more detail. Consider the decay rate o f the state where only y excitations are present, so that p = 0 . In the large n limit
x~.~t = - c tanh (MH t/2 ) ,
F~D2
Yinst = --c t a n h ( M n t / 2 ) .
( 11 )
The instanton action is given by
(12)
Sins t =~C 4 2M n .
We can describe bosonic excitations near the vacuum states by creation operators h* and w* corresponding to motion along the x and y axes. We are going to estimate the transition rates with fermion number non-conservation by two units (we define the fermionic number as F = ( a t a - b t b ) :
Mr, = _ _ (n, n - k i n , n - p ) + + ,
(13)
where In, n - p > + +
=
1
1
=
1
1
× (h*)k(w*) (~-~)a* I - - ) •
× exp{-2[Sinst-nlog(6eSi.stV/-2/n)]}.
(18)
Again, instanton calculation gives exponentially growth with energy rate o f anomalous fermion number non-conservation. The "unitary" bound is violated at EinitiaI ~-) ~ Esoh. Now, if we use the arguments o f refs. [8,9,11,12] we arrive at the conclusion that the rate of fermionic non-conservations becomes unsuppressed at those energies. However, we shall see in a m o m e n t that this conclusion is qualitatively wrong in this Q M model. Consider first this system from a classical point o f view. The form o f the interaction ensures that the initial state with x = c and some excitation along the y axis will never go to the vicinity o f the I - - ) state, independent o f the initial energy. That contradicts
SinstMn/log(6e~
(18).
× (h*)P(w *) (~-P)b*l + + ) , In, n - k > _ _
2~Aferm n
(14)
N o w let us turn to quantum effects. Note that result (18) depends on the coupling p only through the determinant and does not vanish at p = 0 . For this value o f p the quantum hamiltonian o f our system is the sum of two decoupled hamiltonians describing 495
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independent dynamics of x and y coordinates. This means that the rate of fermion number violating transition is given by the product of two factors attributed to the x and y degrees of freedom, /'=rx/~ •
(19)
These factors should be found from the consideration of two one-dimensional problems. For a particular initial state that we choose the system is in the vacuum for x-type excitations, so F~ ~ e x p ( - Sin~)
(20)
(note that the instanton action for the one-dimensional system is smaller by a factor of 2). At the same time, the energy of the y excitation is of the order of the barrier height separating different vacua, so Fy can be of order 1. Hence, the exact solution at p = 0 gives the exponentially suppressed rate, which, however, may be exponentially enhanced [by a factor exp(+Si.st)] in comparison with the low energy limit. The physical reason of this result is clear: in order to get absence of suppression we should excite two degrees of freedom simultaneously. The energy in each degree of freedom should be of the order of the sphaleron mass. The high energy stored in one degree of freedom is of no help. What will happen at small but finite p? If the starting energy is larger than the sphaleron mass, the characteristic time t of the transition will coincide with the time necessary to exite x degrees of freedom to E~ph. The probability of the transition (n, n - k) ~ (n, n - k - 2 ) is of order p 2, so we expect that t ~ M R / p 2. In this case the physics of the transition has nothing to do with instantons at all and the dependence of the rate on the coupling constants is completely different from ( 18 ), though the exponential suppression is absent too. Note, however, that at high energy collisions the time of interaction between different oscillators is small enough. In electroweak theory it can not be bigger than the inverse W mass. So we can imitate high energy collisions by considering the system during a small time interval At~ M ~ ~. The amplitude of the over-barrier transition for E ~ 2Esph c a n be estimated in the framework of perturbation theory as
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14 June 1990
Mf~ = __ (n, 0 [ e x p ( - i H i , t At)In~2, n / 2 ) _ _
~pn/2.
(21)
In other words, we are getting again an exponential suppression of the rate F ~ e x p [ ( - E s p h / M n ) × log(M3n/p)]. We conclude that the absence of suppression can occur only in the case when sufficient energy is stored in the sphaleron negative mode. The divergence of the instanton result at high energies signals the unsuccessful choice of starting with a semiclassical configuration rather than strong coupling. High temperatures provide the possibility to have equipartition of the energy distribution, so p ~ n/2. Of course, the "topological" transitions will be unsuppressed if the temperature of the system T ~ E~ph. 4. We expect to have the same situation in high energy collisions in realistic theories. Here the initial state of two colliding particles is nothing but the excitation of two almost linear oscillators with frequencies ~ / 2 . These oscillators weakly interact with each other during a short time of the collision, transferring energy to other degrees of freedom. It seems highly unprobable that practically all energy of the linear oscillators will go to the degree of freedom corresponding to the unique negative sphaleron mode driving Bnon-conservation, and does not excite at all the infinite number of other oscillators. As in the QM model example we have to excite n ~ E s p h / M w ~ l / a w quanta in this mode. The amplitude of this process is suppressed by the factor ~ e x p [ - (1/Otw)log(1/ Otw) ]. Of course, we can expect an exponential enhancement of the inclusive cross section (say, by a factor exp (Sin~t), as in the quantum mechanical example) taking into account a multiparticle final state, but the exponential suppression of order exp ( - xSinst) (with x some still unknown number of order 1 ) should survive. Again, at high temperatures all degrees of freedom are exited, and the energy stored in the negative sphaleron mode is T/2 according to thermodynamics. When the temperature reaches the sphaleron mass value classical topological transitions are allowed. It is easy to suggest a model for the behaviour of matrix elements which gives unsuppressed processes
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with B-non-conservation at high t e m p e r a t u r e s while preserving the exponential suppression o f the cross section in high energy collisions. It comes in fact from the coherent state representation o f the sphaleron configuration elaborated in ref. [ 15 ]. We can express the rate o f B-non-conservation at high t e m p e r a t u r e s through the matrix elements as
I a m grateful to S.Yu. Khlebnikov, L. McLerran, A. Ringwald, V.A. Rubakov, and P.G. T i n y a k o v for m a n y interesting discussions related to high energy a n o m a l o u s fermionic n u m b e r non-conservation.
r~ph
[ 1] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D 14 (1976) 3432. [2 ] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212. [ 3 ] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36. [4] V.A. Rubakov and A.N. Tavkhelidze, Phys. Lett. B 165 (1985) 109; V.A. Rubakov, Prog. Theor. Phys. 75 (1986) 366; V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Nucl. Phys. B 282 (1987) 700. [ 5 ] M.E. Shaposhnikov, JETP Lett. 44 ( 1986 ) 465; Nucl. Phys. B287 (1987) 757;B 299(1988) 797. [6] V.A. Rubakov, Nucl. Phys. B 256 (1985) 509; J. Amborn and V.A. Rubakov, Nucl. Phys. B 256 (1985) 434. [ 7 ] N.H. Christ, Phys. Rev. D 21 (1980) 1591. [ 8 ] A. Ringwald, preprint DESY 89-074 ( 1989 ). [9] O. Espinosa, preprint CALT-69-1586 (1989). [ 10] P. Arnold and L. McLerran, Phys. Rev. D 37 (1988) 1020. [ 11 ] L. McLerran, A. Vainshtein and M. Voloshin, Minnesota preprint TPI-MINN-89/36-T (1989). [ 12 ] L. McLerran, A. Vainshtein and M. Voloshin, Minnesota preprint TPI-MINN-90/8-T (1990). [13]J. Kripfganz and A. Ringwald, preprint DESY 89-161 (1989). [14IS.Yu. Khlebnikov, V.A. Rubakov and P.G. Tinyakov, Moscow Institute for Nuclear Research preprint (1990). [ 15 ] H. Aoyama and H. Goldberg, Phys. Len. B 188 (1987) 506; H. Goldberg, Phys. Rev. Lett. 62 (1989) 1952.
f ~ [ p h a s e space] (27~)4t~4(pi-Pf) IMn I z
x I-I n ( i ) I-[ [ 1 + n ( f ) ] , i
(22)
f
where n is the Bose or F e r m i distribution. Let us use for simplicity the relativistic phase space formulae and the Boltzmann d i s t r i b u t i o n n = e x p ( - E / T ) instead o f q u a n t u m ones. Then for / 2-\(Nf+Ni)/2 . V Sinst ~
Mn =const." exp ( - Sinst) ~--E-~--)
XO(E-Esph) ,
Nf!Ni! (23)
where v is the v a c u u m expectation value, E is the energy o f the colliding particles, Nf a n d Ni are the number o f bosons in the final and initial states correspondingly, exponential instanton suppression o f the rate at high t e m p e r a t u r e is absent (the factorial dependence is inspired by ref. [ 15 ] ). It disappears after s u m m i n g up final and initialstates. At the same time, for high energy collisions we sum only final states, so the cross section is o f order exp( - Sinst). This is much larger than the rate for transitions at small energies [exp ( -2Sinst ) ] but still small to be observed.
References
497
PHYSICSLETTERSB
14 June 1990
H I G H - E N E R G Y A N O M A L O U S B-NON-CONSERVATION - P H A N T O M OR REALITY? M.E. SHAPOSHNIKOV Institutefor Nuclear Research of the USSR Academy of Sciences, SU-117 312 Moscow, USSR
Received 28 March 1990
I discuss the recent suggestionthat processes with anomalous baryon number non-conservationcan be unsuppressed in high energy collisions. On the example of exactly solvable quantum mechanics I demonstrate that the implementation of the naive euclidean instanton technique gives a qualitatively wrongresult for the rate of the analogueof B-non-conservation.It is clarified how it is possible to have exponentially small cross sections in high energy collisions and an unsuppressed rate of B violation at high temperatures.
1. It is well known that the fermionic number is non-conserved in standard electroweak theory [ 1 ]. The reason is the anomaly in the fermionic current and the complicated vacuum structure in non-abelian gauge theories. Nevertheless, the baryonic and leptonic numbers are good labels for particle degrees of freedom under the usual conditions of low temperatures and densities. To have B-non-conservation, the system must tunnel through the potential barrier, separating vacua with different topological numbers. The height of the barrier coincides with the sphaleron mass Esph --~(3-5.4)Mw/otw [ 2 ]. So B-violation is suppressed by the tremendous semiclassical factor exp(-4n/O~w) [1 ]. This is definitively not the case if the temperature [3] or the density [4] of the system is high enough. Here the system can overcome the barrier classically provided the temperature (or the Fermi momentum for dense media) is compared with the effective height of the barrier. It is needless to say that this issue is important for early cosmology, in particular to the problem of the baryon asymmetry of the universe [3,5]. Moreover, one can expect strong B-non-conservation in decays ofsuperheavy technibaryons [6]. The natural intention is to address the question on the possibility of rapid B-non-conservation in high energy collisions [7 ]. The motivation is obvious: if the centre of mass energy exceeds the sphaleron mass one may naively expect that the exponential suppression characteristic for vacuum-vacuum transitions
will disappear. Interest to this problem was rapidly growing up after the calculation by Ringwald [ 8 ] (see also ref. [ 9 ] ) of the corresponding amplitudes in the instanton approximation. It has been found that this approximation definitely breaks down at high energies corresponding to the sphaleron mass. The main point of calculation takes multiple gauge and Higgs boson production into account, considerably enhancing the cross section of B-violation. The importance of the many-particle final state was previously discussed in ref. [10]. It was speculated in refs. [ 8,9,11,12 ] that the breakdown of the instanton approximation signals the invalidity of the small coupling expansion at high energies and indicates that the weak interactions become strong at X/~~Esph. This implies strong B-non-conservation, multiple W, Z and H production and other phenomena characteristic for strongly coupled theory [ 1 1,12 ]. The breaking of the instanton approximation at high energies is not specific for electroweak theory only. It was shown to take place in a number of twodimensional models of quantum field theory [ 1 1,13,14]. An important result was found in ref. [14] for the 0 ( 3 ) a-model with explicit symmetry breaking. It has been shown there that quantum corrections to the rate become large well before the centre of mass energy becomes equal to the sphaleron energy, namely at V/s~ gwEsph. So the cross section is exponentially small in the region of applicability of instanton calculus.
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
493
Volume 242, number 3,4
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The aim of this paper is to discuss this possibility in the framework of simple quantum mechanical (QM) models. I shall demonstrate that the implementation of the same technique, as was used in electroweak theory, also gives exponentially growth with energy rate of the nonperturbative transitions. However, the exact solution to the problem in QM models with several degrees of freedom keeps the exponential suppression for all initial energies. I conclude, therefore, that the breaking of instanton calculus at high energies signals the unsuccessful choice of a classical configuration to expand about, rather than strong coupling. In other words, strong anomalous B-nonconservation at high energies seems to be very unprobable. Nevertheless, rapid topological transitions at high temperatures take place. 2. Consider first the QM model of one particle in a double well potential. The euclidean action is
S=½~2+U(x),
(1)
U ( x ) = ~ ( x 2 - c 2 ) 2.
To explore the energy dependence of the amplitude we can address the question about the probability of transition from an exited state with n quanta in the left well to a state with the same number of quanta in the right well. Following refs. [8,9] we consider the euclidean functional integral for the multiparticle Green function:
G(tl .... , tzn) = f D[x]
exp(-S)(x(t,)...x(t2,)).
(2)
With the use of the instanton solution
xi.~t=ctanh(Mnt/2),
M2=22c 2 ,
(3)
we get in the lowest approximation
G=O f
dto ~
i=1
Sinst = 2C2MH ,
xi,st(ti-to)exp(-Si,st)
, (4)
where D stands for the gaussian functional integral around the instanton without zero modes, to is the position of the instanton. Zero mode normalization is included to D too. We can get the matrix element of transition Mn by (i) calculating the Fourier transform of (4) with exp(iZogit~), (ii) making an analyt494
14 June 1990
ical continuation to Minkowski time, to~i~oi, (iii) putting the Green function to the mass shell with the help of reduction formulae and taking into account statistical factors coming from the normalization of states. The "leg" amputation is given by ( 2 M n ) I / 2 x (e~- MH). The final result for the probability of transition is / " = O 2 ( 1 2 S i n s t ) 2n (n!) 2 e x p ( - 2Sinst) •
(5)
In the large n limit we get
F~-DZ2nexp{-2[Si,st-nlog(12eSi,~t/n)]). n
(6)
We see that in quantum mechanics, as in field theory, the instanton calculation gives exponentially growth with energy E (recall that n = E/Mn) "cross section" of anomalous fermion number non-conservation, hitting the "unitary" bound at E~SinstMH/ log( 12e) -- 1.5Esph, where = L,K sph' ~t,l~21wt21"4"2His the height of the barrier separating different vacua (sphaleron mass). It can be easily shown by calculating the corrections to eq. (2) that the instanton approximation breaks down at n -- (Sinst) !/2 i.e. at energies E~_ Esph/ (ref. [14] ). The explanation is obvious: the instanton configuration is a bad starting point to expand about at E~-E~ph. For a correct semiclassical treatment we should use a periodic euclidean configuration given by the equation dx
v/~[ U ( x ) _ E ] =t ,
(7)
explicitly involving the initial energy. This solution differs drastically from the initial instanton, even the boundary conditions are to be changed. Nevertheless, the result is qualitatively correct at energies ~ Esph. The reason is simple: for one degree of freedom all the energy is concentrated in a "right direction" coinciding with the negative mode of the sphaleron. Let us now go to a bit more complicated example with two degrees of freedom where the initial energy can be attributed to the "orthogonal direction". 3. Consider the QM model with two degrees of freedom x and y with the euclidean action
Volume 242, number 3,4
PHYSICS LETTERS B
With the use of the expression for normalizable zero fermionic modes
& = ½ ~ + iX(x ~ _ c ~) ~+ i p = + i X ( y ~ - c ~)
1
-{- ~ C 4 P [ ( X 2 - - C 2 ) 2 - ( y z - c 2 ) 2 1 2 .
(8)
It has four degenerate vacua at the classical level:
~o = N 1/2 [cosh (Mn t12 ) ] -4flMn ,
( 15 )
we get for the Green function describing the transition the following:
I++ >={x=c,y=c},
G = D ~ dto ~0 (t2n+ 1 - t o ) ~o(t2n+2 --to)
I + - > ={x=c, y = - c } ,
I-+ >={x=-c,y=c}, I--
14 June 1990
2n
) ={x=-c,y=-c}.
(9)
For p < 2 the sphaleron configurations are nothing but the points ( x = 0 , y = + c ) , ( y = 0 , x=_+c), E~ph= 1 Rat2 ~,* u~p 2 /~ 1 + p / 2 2 ) . Let us add fermions to the theory with the interaction
S f = f ( a * a - b'b) ( x + y) ,
( 1O)
× ~ Xinst(ti--to), i=1
(16)
in obvious notation. Using the same procedure as the one-dimensional Q M model we obtain the probability o f exclusive transition from the initial state i with fermion number non-conservation:
F=AfermD2 (6Sinst p ! ( n -X/2)2n p ) ! exp ( - 2Sinst)
(17)
where a, a*, b and b t are anticommuting operators, f is the Yukawa coupling. We shall have fermionic n u m b e r non-conservation together with a transition between the vacuum I + + ) and the vacuum I - - ). This transition is described by the instanton configuration
where Arerm is a factor coming from fermionic zero modes. It is not essential in what follows. Let us analyze this equation in more detail. Consider the decay rate o f the state where only y excitations are present, so that p = 0 . In the large n limit
x~.~t = - c tanh (MH t/2 ) ,
F~D2
Yinst = --c t a n h ( M n t / 2 ) .
( 11 )
The instanton action is given by
(12)
Sins t =~C 4 2M n .
We can describe bosonic excitations near the vacuum states by creation operators h* and w* corresponding to motion along the x and y axes. We are going to estimate the transition rates with fermion number non-conservation by two units (we define the fermionic number as F = ( a t a - b t b ) :
Mr, = _ _ (n, n - k i n , n - p ) + + ,
(13)
where In, n - p > + +
=
1
1
=
1
1
× (h*)k(w*) (~-~)a* I - - ) •
× exp{-2[Sinst-nlog(6eSi.stV/-2/n)]}.
(18)
Again, instanton calculation gives exponentially growth with energy rate o f anomalous fermion number non-conservation. The "unitary" bound is violated at EinitiaI ~-) ~ Esoh. Now, if we use the arguments o f refs. [8,9,11,12] we arrive at the conclusion that the rate of fermionic non-conservations becomes unsuppressed at those energies. However, we shall see in a m o m e n t that this conclusion is qualitatively wrong in this Q M model. Consider first this system from a classical point o f view. The form o f the interaction ensures that the initial state with x = c and some excitation along the y axis will never go to the vicinity o f the I - - ) state, independent o f the initial energy. That contradicts
SinstMn/log(6e~
(18).
× (h*)P(w *) (~-P)b*l + + ) , In, n - k > _ _
2~Aferm n
(14)
N o w let us turn to quantum effects. Note that result (18) depends on the coupling p only through the determinant and does not vanish at p = 0 . For this value o f p the quantum hamiltonian o f our system is the sum of two decoupled hamiltonians describing 495
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PHYSICS LETTERS B
independent dynamics of x and y coordinates. This means that the rate of fermion number violating transition is given by the product of two factors attributed to the x and y degrees of freedom, /'=rx/~ •
(19)
These factors should be found from the consideration of two one-dimensional problems. For a particular initial state that we choose the system is in the vacuum for x-type excitations, so F~ ~ e x p ( - Sin~)
(20)
(note that the instanton action for the one-dimensional system is smaller by a factor of 2). At the same time, the energy of the y excitation is of the order of the barrier height separating different vacua, so Fy can be of order 1. Hence, the exact solution at p = 0 gives the exponentially suppressed rate, which, however, may be exponentially enhanced [by a factor exp(+Si.st)] in comparison with the low energy limit. The physical reason of this result is clear: in order to get absence of suppression we should excite two degrees of freedom simultaneously. The energy in each degree of freedom should be of the order of the sphaleron mass. The high energy stored in one degree of freedom is of no help. What will happen at small but finite p? If the starting energy is larger than the sphaleron mass, the characteristic time t of the transition will coincide with the time necessary to exite x degrees of freedom to E~ph. The probability of the transition (n, n - k) ~ (n, n - k - 2 ) is of order p 2, so we expect that t ~ M R / p 2. In this case the physics of the transition has nothing to do with instantons at all and the dependence of the rate on the coupling constants is completely different from ( 18 ), though the exponential suppression is absent too. Note, however, that at high energy collisions the time of interaction between different oscillators is small enough. In electroweak theory it can not be bigger than the inverse W mass. So we can imitate high energy collisions by considering the system during a small time interval At~ M ~ ~. The amplitude of the over-barrier transition for E ~ 2Esph c a n be estimated in the framework of perturbation theory as
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14 June 1990
Mf~ = __ (n, 0 [ e x p ( - i H i , t At)In~2, n / 2 ) _ _
~pn/2.
(21)
In other words, we are getting again an exponential suppression of the rate F ~ e x p [ ( - E s p h / M n ) × log(M3n/p)]. We conclude that the absence of suppression can occur only in the case when sufficient energy is stored in the sphaleron negative mode. The divergence of the instanton result at high energies signals the unsuccessful choice of starting with a semiclassical configuration rather than strong coupling. High temperatures provide the possibility to have equipartition of the energy distribution, so p ~ n/2. Of course, the "topological" transitions will be unsuppressed if the temperature of the system T ~ E~ph. 4. We expect to have the same situation in high energy collisions in realistic theories. Here the initial state of two colliding particles is nothing but the excitation of two almost linear oscillators with frequencies ~ / 2 . These oscillators weakly interact with each other during a short time of the collision, transferring energy to other degrees of freedom. It seems highly unprobable that practically all energy of the linear oscillators will go to the degree of freedom corresponding to the unique negative sphaleron mode driving Bnon-conservation, and does not excite at all the infinite number of other oscillators. As in the QM model example we have to excite n ~ E s p h / M w ~ l / a w quanta in this mode. The amplitude of this process is suppressed by the factor ~ e x p [ - (1/Otw)log(1/ Otw) ]. Of course, we can expect an exponential enhancement of the inclusive cross section (say, by a factor exp (Sin~t), as in the quantum mechanical example) taking into account a multiparticle final state, but the exponential suppression of order exp ( - xSinst) (with x some still unknown number of order 1 ) should survive. Again, at high temperatures all degrees of freedom are exited, and the energy stored in the negative sphaleron mode is T/2 according to thermodynamics. When the temperature reaches the sphaleron mass value classical topological transitions are allowed. It is easy to suggest a model for the behaviour of matrix elements which gives unsuppressed processes
Volume 242, number 3,4
PHYSICS LETTERS B
14 June 1990
with B-non-conservation at high t e m p e r a t u r e s while preserving the exponential suppression o f the cross section in high energy collisions. It comes in fact from the coherent state representation o f the sphaleron configuration elaborated in ref. [ 15 ]. We can express the rate o f B-non-conservation at high t e m p e r a t u r e s through the matrix elements as
I a m grateful to S.Yu. Khlebnikov, L. McLerran, A. Ringwald, V.A. Rubakov, and P.G. T i n y a k o v for m a n y interesting discussions related to high energy a n o m a l o u s fermionic n u m b e r non-conservation.
r~ph
[ 1] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D 14 (1976) 3432. [2 ] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212. [ 3 ] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36. [4] V.A. Rubakov and A.N. Tavkhelidze, Phys. Lett. B 165 (1985) 109; V.A. Rubakov, Prog. Theor. Phys. 75 (1986) 366; V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Nucl. Phys. B 282 (1987) 700. [ 5 ] M.E. Shaposhnikov, JETP Lett. 44 ( 1986 ) 465; Nucl. Phys. B287 (1987) 757;B 299(1988) 797. [6] V.A. Rubakov, Nucl. Phys. B 256 (1985) 509; J. Amborn and V.A. Rubakov, Nucl. Phys. B 256 (1985) 434. [ 7 ] N.H. Christ, Phys. Rev. D 21 (1980) 1591. [ 8 ] A. Ringwald, preprint DESY 89-074 ( 1989 ). [9] O. Espinosa, preprint CALT-69-1586 (1989). [ 10] P. Arnold and L. McLerran, Phys. Rev. D 37 (1988) 1020. [ 11 ] L. McLerran, A. Vainshtein and M. Voloshin, Minnesota preprint TPI-MINN-89/36-T (1989). [ 12 ] L. McLerran, A. Vainshtein and M. Voloshin, Minnesota preprint TPI-MINN-90/8-T (1990). [13]J. Kripfganz and A. Ringwald, preprint DESY 89-161 (1989). [14IS.Yu. Khlebnikov, V.A. Rubakov and P.G. Tinyakov, Moscow Institute for Nuclear Research preprint (1990). [ 15 ] H. Aoyama and H. Goldberg, Phys. Len. B 188 (1987) 506; H. Goldberg, Phys. Rev. Lett. 62 (1989) 1952.
f ~ [ p h a s e space] (27~)4t~4(pi-Pf) IMn I z
x I-I n ( i ) I-[ [ 1 + n ( f ) ] , i
(22)
f
where n is the Bose or F e r m i distribution. Let us use for simplicity the relativistic phase space formulae and the Boltzmann d i s t r i b u t i o n n = e x p ( - E / T ) instead o f q u a n t u m ones. Then for / 2-\(Nf+Ni)/2 . V Sinst ~
Mn =const." exp ( - Sinst) ~--E-~--)
XO(E-Esph) ,
Nf!Ni! (23)
where v is the v a c u u m expectation value, E is the energy o f the colliding particles, Nf a n d Ni are the number o f bosons in the final and initial states correspondingly, exponential instanton suppression o f the rate at high t e m p e r a t u r e is absent (the factorial dependence is inspired by ref. [ 15 ] ). It disappears after s u m m i n g up final and initialstates. At the same time, for high energy collisions we sum only final states, so the cross section is o f order exp( - Sinst). This is much larger than the rate for transitions at small energies [exp ( -2Sinst ) ] but still small to be observed.
References
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