International Journal of Modern Physics A Vol. 16, Suppl. 1C (2001) 1238-1240 © World Scientific Publishing Company
THE DISPERSIVE APPROACH TO ELECTROWEAK PROCESSES IN THE BACKGROUND MAGNETIC FIELD
GUEY-LIN LIN
Institute of Physics, National Chiao-Tung University, 1001 Ta-Hsueh Rd Hsinchu 300, Taiwan
We propose a new method t o compute amplitudes of electroweak processes in the strong background magnetic field, using 7 -+ e+e~" as an example. We show t h a t the moments
of 7 —>• e+e ~ width are proportional to the derivatives of photon polarization function
at the zero energy. Hence, the pair-production width can be easily calculated from the latter by the inverse Mellin transform. The prospects of our approach are commented. The electroweak phenomena associated with an intensive background magnetic field are rather rich. Under a background magnetic field, a physical photon can decay into an e+e~~ pair or split into two photons. Such processes are relevant to the attenuation of gamma-rays from pulsars1'2. Similarly, with a sufficient energy, a neutrino can go through the decays v —• i/e+e~~3 and u —• 1/74. For processes without charged fermions in the final state, such as 7 -» 77 or v -> 1/7, their decay widths can be expressed as asymptotic series in B for B < Bc. However, such
asymptotic expansions are not possible for v -* ve+e~ or 7 -> e+e~ since the wave
functions of final-state fermions are non-analytic with respect to the magnetic field strength at B = 0.
Previously, the photon absorption coefficients due to 7 -> e+e~~ were computed in two different ways. One either directly squares the 7 -» e+e"~ amplitude using the exact electron (positron) wave function in the background magnetic field5, or applies the optical theorem on the photon polarization function nM„6. In both approaches, the results are valid only for B < Bc and o;sin# >> 2rae, where u is the photon energy and 6 is the angle between the magnetic-field direction and the direction of photon propagation. It has been pointed out7 that a correct description of 7 -> e+e~ near the pair-production threshold a>sin0 « 2rae is crucial for astrophysical applications. A numerical study taking into account the threshold behavior of 7 -> e+e~ was also carried out7. In this work, we re-examine the previous analytic approaches to 7 -* e+e~ 5'6, clarifying their implicit assumptions which lead to incorrect threshold behavior for the above decay.
Let us follow Ref. 6 which begins with the proper-time representation8'9 of pho-ton polarization function nM„ in the background magnetic field:
. 3 D poo f+1 - ? T ^ 2 / ds / M e - * * » [ f a V " tyfcOJVb v^r Jo J-1 («f5||/iy - Q\\tAQ\\v)N\\ + (q±9±^ - q±pQjLV)N±]
e - ^ ( l - t /
2) ( g V ^ ^ ) }
5(1)
where
2 1-V2 2 C0B(ZV) - COB(Z) 2 /0N * > = » » . - - 4 - I I " 2,sin(.) ^ ( 2 ) iW<?) 1238The Dispersive Approach to Electroweak Processes 1239 with z = eBs, and JVO,||,_L trigonometric functions of z and v. Here || and ± are defined relative to the magnetic-field direction.
The photon dispersion relation is given by q2 + Rell^x = 0> where II^x =
6f j_IWejf ± with ejf and e± respectively the the photon polarization vectors par-allel and perpendicular to the plane spanned by the photon momentum q and the magnetic field B. The imaginary part of II^x is related to photon absorption coef-ficients K\\$± (i.e. the width of 7 -» e+e~~) via K\\,± = Imn^i/a;, with u the photon energy. The authors of Ref. 6 analyzed the functions II||}x in the limit u sin 0 >• 2rae and B < Bc. They found K\\t± = |asin0(eB/me)T\\i ±(\), with
r
>- w = 3 r I! ** - -
!
»-' [0 - r
2
»- f I
+
H
K
«> (jih) •
(3) where A = §(eB/ra2)(u;/rae)sin0 and if2/3 is the modified Bessel function. Com-pared to the numerical study7, this result is accurate at the higher energy with £ = u2 sin2 0Bc/2m2B > 103. However its low energy prediction is problematic. One would expect that T^t± (A) vanishes for w sin 0 below the pair productionthresh-old. On the other hand, for A < 1, Tlu± -> (3/2)1/2 . ( l / 2 , l / 4 ) e -4/ \ which does not behave like a step function. Furthermore T\\t± is a smooth function of A, while
in actual situation it should contain infinite many sawtooth absorption edges cor-responding to higher Landau levels reachable by the increasing photon energies. These discrepancies might have to do with the approximation made in Ref. 6 where only the small-s contribution in Eq. (1) is taken into account. However, due to the highly oscillatory behavior of the integrand, it remains unclear how to evaluate the large-s contribution.
Recently, we have developed a technique to deal with the external-field problem, using the analytic properties of physical amplitudes10. We observe that once the amplitude of a physical process is known in the small momentum (energy) regime, its behavior at arbitrary momentum (energy) is completely determined by the inverse Mellin transform. For the current problem, we have10
M^
n
»-)u.=^r*-"-'-(^
w
^)-where y = M2 ±/u2 with M\\}± the threshold energies of pair productions5'11 given by M2 sin2 0 = 4ra2 and M\ sin2 0 = m2 (1 + 1/1 + 2B/B^\ . One notes that the imaginary part of n||}x(a;2) vanishes for the range 0 < u2 < M2 ±. This property
has been verified in the previous works5'11. Therefore one can effectively set the integration range of Eq. (4) as from y = 0 to y = 00. It is then obvious that the derivatives of n ^ x at the zero energy are proportional to the Mellin transform of /C||}x -y~lf2 = >S||}±*kV-fl%±« Once the l.h.s. of Eq. (4) is calculated, the absorption coefficients K^± can be determined by the inverse Mellin transform.
The l.h.s of Eq. (4) is calculated with a rotation of integration contour s -» —is, which is permissible only for u) below the pair-production threshold. By this rotation, the phase factor exp(—isfa) in Eq. (1) turns into the more well-behaved factor exp(—s0o) where (/>Q is obtained from <f>o by the replacement z —• — iz (z =
eBs). Furthermore, the trigonometric functions in the integrand iV0)||}x also turn into the hyperbolic function of z and v. Due to the presence of exp(—S(J>Q) and
1240 Guey-Lin Lin
the assumption of a sub-critical magnetic field B < Bc, one may obtain a
first-approximation for (d/da;2)nII||j_L|w2=o by disregarding the large-s contribution in
the integral of Eq. (1) and its derivatives. This amounts to approximating, for example, cosh(^) there by the power series 1JrZ2j2 H . With this approximation,
we arrive at
1 / dn \ i 2am2e (B2 sin2 0\ n T(3n - l)T2(2n)
n! \d(w2)n ll»J7 L = o 7T V 3 ^ ? ^ i / T(n)r(4n)
where the neglected terms axe suppressed by the factor (B/Bc)2. Combining
Eqs. (4) and (5), one obtains the photon absorption coefficients K\\t± by the
in-verse Mellin transform10:
_ a m ; r +ioo+a
, ,,\
2»r(3s)r
2(2s) 1 6s+ 1
« j . = -ioo+a V ' r ( s ) r ( 4 s ) 3 s - 1 4 s + 1 ' 2am2 1 r+ioo+a 2 8r ( 3 s ) r2( 2 s ) 1 3 s + 1 »*w 1 + ^ 1 + 2 B / BC y _i 0 0 + 0 ( ' T(s)r(4s) 3s - 1 * 4s + 1' (6) where a is any real number greater than 1/3; while A' = (u sinGB/^/3meBc) andA = A • (1 + y/1 + 2B/Bc)/2. It can be shown rigorously that1 0 K\\t± computed
in this way are equivalent to results of Tsai and Erber given in Eq. (3), except on some trivial kinematic factors.
The work on improving Eq. (5) and consequently the threshold behavior of K^t±
is in progress12. We have found that the large-s contribution which is disregarded in
the first-approximation becomes important in the higher derivatives of II||}_L- Taking
into account this contribution is crucial to obtain correct threshold behaviors for
*II.-L-This work is supported in part by the National Science Council of Taiwan under the grant number NSC89-2112-M009-035.
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