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Elementary examples of baryon number generation
Volume
81 B, number
2
PHYSICS
LETTERS
12 February
1979
ELEMENTARYEXAMPLESOFBARYONNUMBERGENERATION* D. TOUSSAINT 1 and Frank WILCZEK Joseph Henry Laboratories, Princeton University,Princeton, NJO8540, USA Received
29 September
1978
Elementary examples are provided of reaction rate matrices satisfying the constraints of unitarity and CPT invariance, which can lead a system with baryon number initially zero to evolve toward non-zero baryon number. The importance of the unitarity constraint is emphasized.
In a previous paper [I] we argued that, in the presence of baryon-number, C, and T violating interactions, systems out of thermal equilibrium will in general evolve into conditions with non-zero net baryon number, even if they begin with zero baryon number. We exemplified this by the decay of hypothetical heavy “K” -like” mesons M, M which could have unequal rates for decays like M + p+ + e-, M + p + e+. Then a symmetric distribution of M and M will decay into a final state with non-zero baryon number. It is of interest to know whether an appreciable baryon number could also be generated by collisions without mixing, since the M-system (like the K” SYStern) might seem to be very special. Some concrete calculations along these lines have been attempted recently [2,3] . (See also refs. [ 1,4] for general discussions of baryon-number generation.) These calculations are very difficult to survey since they were made in the context of specific, complicated models of baryon-number violation. (In fact we argued in ref. [ 1] that the reported result of the calculation in ref. [2] cannot be correct, since it violates a general theorem.) It seems worthwhile, therefore, to provide some explicit examples of reaction matrices satisfying the relevant general principles which can lead to baryon-number generation. The important principles which must be respected * Research
partially supported by Department of Energy under contract EY-76C02-3072 and NSF contract PHY 78-01221. 1 Permanent address: University of California, Santa Barbara.
238
in such an example are the CPT theorem and conservation of probability (unitarity). The importance of unitarity, which might not be immediately obvious, is illustrated by the following consideration: The CPT theorem relates the reaction p + p -+ j7 t p only to itself, SOthat p + p + p + p could have a different rate from -p + p -+ p + p. On this basis it would seem that an assembly of p and fi would evolve asymmetrically. However, we show in example 1 below that this conclusion is false when the unitarity constraint is imposed. From a deeper point of view, the unitarity constraint is the key to proving that thermal equilibrium is approached (H-theorem) even in a system with T-violation [S] (see also ref. [ 11). In thermal equilibrium particle-antiparticle symmetry is guaranteed since the Boltzmann distribution depends only on masses, which are equal for particle and antiparticle by the CPT theorem. Even for understanding kinetically how equilibrium can arise without Tinvariance the unitdrity constraint is essential. In the examples we shall severely idealize our problem by considering a very restricted number of channels (one might think of S-wave scattering at a fixed energy). Nothing essential is lost this way, and some very cumbersome notation is avoided. Example I: a single species b. This is actually a sort of “anti-example”, intended to show the importance of the unitarity constraint. Let us make a reaction matrix T = -i(S1) as follows. Call bb, 66, b6 combinations 1,2,3, respectively. We then define Tij
as the amplitude for combination i to scatter into combination j (similarly Sii). We define J to be the matrix
J= i
0 1 0
1 0 0
0 0 , 1 1
21T,,12 •tlT,,12 + 1T,,12 - 21Tz112 _ IT31l2 - IT,,12 = A. Now CPTrequires (1)
so J2 = 1. Then the CPT constraint JSJ = ST
(CPT) .
(3)
A simple way of insuring that S be unitary is to write S=eti,
(4)
JHJ = HT JHJ = H
Then the CPT, CP constraints
are, (5)
(CP) .
where the last equality follows from unitarity 1+ TT+ = T+T).
with CPT is
(9)
(SS+ =
Example 2: Two species, baiyon number generation. We can perform a very similar analysis with two species p, b and four combinations pp, pb, Pp, pb. The new J-matrix is
J=
0
I\o
1
o\
0 0 0 1 1
(6)
Thus the form of H consistent
+ 1T1212+ IT1312
= 2C(TT+)II - (T+Tlll )=O,
/o
(CPT) ,
ITs212 = IT1312, lT2812 = ITS112, so
- IT,, I2 - 1T21I2 - IT#}
(2)
(CP) .
with H hermitean. respectively
A=2{1Tl,12
on the S-matrix is
Note that CP (or T, assuming CPT) gives JSJ = S
12 February 1979
PHYSICS LETTERS
Volume 81B, number 2
0 1
0 0
0 1’ 01
(10)
and His of the form iQ
(7)
a!
P
Y\
(11) where the Latin letters represent real numbers, the Greek letters complex numbers, and the overbar complex conjugation. If CP is assumed in addition then o, p are real. Now we have constructed a unitary, CPT-invariant S-matrix. One may readily calculate whether the transition 11 + 11 is equal to the rate for fl+ 11. One finds it is not. Explicitly, to third order in the entries of H (assumed small) the two rates differ by a (CPviolating) term lT1212 - 1T21l2 = 21rnd2
.
(8)
However, this--does not lead to an --asymmetry! Six reactions bb + bb, bb + bb, bb + bb, bb + bF, bb-+ bb, bb + bb, must all be considered to see if there is an asymmetry. When one adds them up, the rates of change of b and bare found to be identical (starting from a symmetric distribution). In fact, if we start with equal amounts of b and b the rate of change of b-b will be proportional to
We find to third order ITI
t lTs212 - IT1412 - ITSal = 8Imfiys.
(12)
That is, the combined rates for pp + bb, pp + bb do not equal the combined rates for pp + I%,pP + bb. Thus an initially symmetric distribution of p and p with no b orb will produce an asymmetric state of b and b, which is what we wished to exemplify. Remarks. (1) The baryon number generation eq. (12) is third order in the interactions generally and second order in baryon violating interactions. A moment’s thought shows that this result is general in our analysis. We may see the same thing in a different and interesting way by thinking about Feynman graphs. We are looking (at least) for a different rate for some baryon-number violating --interaction AB -+ CD and the antiparticle reaction AB + CD. These two reactions 239
Volume
81 B,
number 2
PHYSICS LETTERS
in Feynman graphs are related by changing the direction of the lines, which amounts to complex conjugation except for the ie prescription in loop graphs. Thus the lowest order in which a difference between AB + CD and AB + CD can arise is from interference between Born and I-ioop graphs, both violating baryon number. Furthermore, only the absorptive part of the loop graph matters, so the rate goes like the square of cross sections of real (not virtual) baryon-violating processes. (2) The main possible physical interest of the foregoing is of course its possible relevance to the question of cosmological baryon-antibaryon asymmetry. We have recently discussed this in general terms [l] . Our purpose in this note is only to show that our
240
12 February 1919
previous emphasis on decays was not essential, and to exhibit the essence of baryon number generation by collisions without commitment to detailed models. We thank S. Weinberg for a stimulating
discussion.
References [l] [2] [3] [4] [S]
D. Toussaint, S. Treiman, F. Wilczek and A. Zee, Matter -antimatter accounting..., Princeton preprint (1978). H. Yoshimura, Phys. Rev. Lett. 41 (1978) 381. A. Ignatiev, N. Krasnikov, V. Kuzmin and A. Tavkhelidze, Phys. Lett. 76B (1978) 436. S. Dimopoulous and L. Susskind, On the baryon number of the universe, SLAC-PUB-2126 (1978). E. Stueckelberg, Helv. Phys. Acta 25 (1952) 577.
81 B, number
2
PHYSICS
LETTERS
12 February
1979
ELEMENTARYEXAMPLESOFBARYONNUMBERGENERATION* D. TOUSSAINT 1 and Frank WILCZEK Joseph Henry Laboratories, Princeton University,Princeton, NJO8540, USA Received
29 September
1978
Elementary examples are provided of reaction rate matrices satisfying the constraints of unitarity and CPT invariance, which can lead a system with baryon number initially zero to evolve toward non-zero baryon number. The importance of the unitarity constraint is emphasized.
In a previous paper [I] we argued that, in the presence of baryon-number, C, and T violating interactions, systems out of thermal equilibrium will in general evolve into conditions with non-zero net baryon number, even if they begin with zero baryon number. We exemplified this by the decay of hypothetical heavy “K” -like” mesons M, M which could have unequal rates for decays like M + p+ + e-, M + p + e+. Then a symmetric distribution of M and M will decay into a final state with non-zero baryon number. It is of interest to know whether an appreciable baryon number could also be generated by collisions without mixing, since the M-system (like the K” SYStern) might seem to be very special. Some concrete calculations along these lines have been attempted recently [2,3] . (See also refs. [ 1,4] for general discussions of baryon-number generation.) These calculations are very difficult to survey since they were made in the context of specific, complicated models of baryon-number violation. (In fact we argued in ref. [ 1] that the reported result of the calculation in ref. [2] cannot be correct, since it violates a general theorem.) It seems worthwhile, therefore, to provide some explicit examples of reaction matrices satisfying the relevant general principles which can lead to baryon-number generation. The important principles which must be respected * Research
partially supported by Department of Energy under contract EY-76C02-3072 and NSF contract PHY 78-01221. 1 Permanent address: University of California, Santa Barbara.
238
in such an example are the CPT theorem and conservation of probability (unitarity). The importance of unitarity, which might not be immediately obvious, is illustrated by the following consideration: The CPT theorem relates the reaction p + p -+ j7 t p only to itself, SOthat p + p + p + p could have a different rate from -p + p -+ p + p. On this basis it would seem that an assembly of p and fi would evolve asymmetrically. However, we show in example 1 below that this conclusion is false when the unitarity constraint is imposed. From a deeper point of view, the unitarity constraint is the key to proving that thermal equilibrium is approached (H-theorem) even in a system with T-violation [S] (see also ref. [ 11). In thermal equilibrium particle-antiparticle symmetry is guaranteed since the Boltzmann distribution depends only on masses, which are equal for particle and antiparticle by the CPT theorem. Even for understanding kinetically how equilibrium can arise without Tinvariance the unitdrity constraint is essential. In the examples we shall severely idealize our problem by considering a very restricted number of channels (one might think of S-wave scattering at a fixed energy). Nothing essential is lost this way, and some very cumbersome notation is avoided. Example I: a single species b. This is actually a sort of “anti-example”, intended to show the importance of the unitarity constraint. Let us make a reaction matrix T = -i(S1) as follows. Call bb, 66, b6 combinations 1,2,3, respectively. We then define Tij
as the amplitude for combination i to scatter into combination j (similarly Sii). We define J to be the matrix
J= i
0 1 0
1 0 0
0 0 , 1 1
21T,,12 •tlT,,12 + 1T,,12 - 21Tz112 _ IT31l2 - IT,,12 = A. Now CPTrequires (1)
so J2 = 1. Then the CPT constraint JSJ = ST
(CPT) .
(3)
A simple way of insuring that S be unitary is to write S=eti,
(4)
JHJ = HT JHJ = H
Then the CPT, CP constraints
are, (5)
(CP) .
where the last equality follows from unitarity 1+ TT+ = T+T).
with CPT is
(9)
(SS+ =
Example 2: Two species, baiyon number generation. We can perform a very similar analysis with two species p, b and four combinations pp, pb, Pp, pb. The new J-matrix is
J=
0
I\o
1
o\
0 0 0 1 1
(6)
Thus the form of H consistent
+ 1T1212+ IT1312
= 2C(TT+)II - (T+Tlll )=O,
/o
(CPT) ,
ITs212 = IT1312, lT2812 = ITS112, so
- IT,, I2 - 1T21I2 - IT#}
(2)
(CP) .
with H hermitean. respectively
A=2{1Tl,12
on the S-matrix is
Note that CP (or T, assuming CPT) gives JSJ = S
12 February 1979
PHYSICS LETTERS
Volume 81B, number 2
0 1
0 0
0 1’ 01
(10)
and His of the form iQ
(7)
a!
P
Y\
(11) where the Latin letters represent real numbers, the Greek letters complex numbers, and the overbar complex conjugation. If CP is assumed in addition then o, p are real. Now we have constructed a unitary, CPT-invariant S-matrix. One may readily calculate whether the transition 11 + 11 is equal to the rate for fl+ 11. One finds it is not. Explicitly, to third order in the entries of H (assumed small) the two rates differ by a (CPviolating) term lT1212 - 1T21l2 = 21rnd2
.
(8)
However, this--does not lead to an --asymmetry! Six reactions bb + bb, bb + bb, bb + bb, bb + bF, bb-+ bb, bb + bb, must all be considered to see if there is an asymmetry. When one adds them up, the rates of change of b and bare found to be identical (starting from a symmetric distribution). In fact, if we start with equal amounts of b and b the rate of change of b-b will be proportional to
We find to third order ITI
t lTs212 - IT1412 - ITSal = 8Imfiys.
(12)
That is, the combined rates for pp + bb, pp + bb do not equal the combined rates for pp + I%,pP + bb. Thus an initially symmetric distribution of p and p with no b orb will produce an asymmetric state of b and b, which is what we wished to exemplify. Remarks. (1) The baryon number generation eq. (12) is third order in the interactions generally and second order in baryon violating interactions. A moment’s thought shows that this result is general in our analysis. We may see the same thing in a different and interesting way by thinking about Feynman graphs. We are looking (at least) for a different rate for some baryon-number violating --interaction AB -+ CD and the antiparticle reaction AB + CD. These two reactions 239
Volume
81 B,
number 2
PHYSICS LETTERS
in Feynman graphs are related by changing the direction of the lines, which amounts to complex conjugation except for the ie prescription in loop graphs. Thus the lowest order in which a difference between AB + CD and AB + CD can arise is from interference between Born and I-ioop graphs, both violating baryon number. Furthermore, only the absorptive part of the loop graph matters, so the rate goes like the square of cross sections of real (not virtual) baryon-violating processes. (2) The main possible physical interest of the foregoing is of course its possible relevance to the question of cosmological baryon-antibaryon asymmetry. We have recently discussed this in general terms [l] . Our purpose in this note is only to show that our
240
12 February 1919
previous emphasis on decays was not essential, and to exhibit the essence of baryon number generation by collisions without commitment to detailed models. We thank S. Weinberg for a stimulating
discussion.
References [l] [2] [3] [4] [S]
D. Toussaint, S. Treiman, F. Wilczek and A. Zee, Matter -antimatter accounting..., Princeton preprint (1978). H. Yoshimura, Phys. Rev. Lett. 41 (1978) 381. A. Ignatiev, N. Krasnikov, V. Kuzmin and A. Tavkhelidze, Phys. Lett. 76B (1978) 436. S. Dimopoulous and L. Susskind, On the baryon number of the universe, SLAC-PUB-2126 (1978). E. Stueckelberg, Helv. Phys. Acta 25 (1952) 577.