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Light cone structure of the current two-point function and asymptotic behaviour in momentum space
Volume 52, number 3
PHYSICS LETTERS
LIGHT CONE STRUCTURE AND ASYMPTOTIC
14 October 1974
OF THE CURRENT BEHAVIOUR
TWO-POINT FUNCTION
IN MOMENTUM
S P A C E ¢~
M. MAGG
Institut fiir TheoretischePhysik, TechnischeHochschuleAachen, FR Germany Received 19 September 1974 We prove the correspondence between the degree of singularity of a two point function on the light cone and the asymptotic behaviour of its Fourier transform. The result is relevant for e+-e - annihilation. Contrary to deep inelastic e-p scattering [e.g., 1 ] recent experiments [2] seem to indicate that e+-e annihilation requires a stronger singular structure than the canonical one for the current commutator. For this interpretation it is crucial that in both processes a) one-photon exchange dominates b) asymptotic behaviour in m o m e n t u m space can be uniquely translated into space-time singularities of the current commutator on the light cone. In the following we shall assume the validity of point a). Regarding the Bjorken limit in e-p scattering a complete discussion of point b) is given in ref. [3]. As the preliminary experiments with e+-e - annihilation seem to contradict many fashionable theoretical models [4] it may be worth-while to investigate the relation between the current two-point function in configuration space and the asymptotic behaviour of otot(s) independent of any special model, using only general principles of quantum field theory like Lorentz-invariance, locality and positivity [5]. The total cross section for e+-e - annihilation into hadrons is proportional to the Fourier transform of the following distribution T:
(O[(ju(x),iv(Y)] lO) =(~xx~ ~x -guvff])T(x-y)"
(1)
T is a tempered, Lorentz-invariant, uneven distribution on the configuration space R 4. It is the antisymmetric part of the positive definite Wightman function (0 Iju(x)ju(y)l 0). Any distribution of this type can be represented by a uniquely determined one dimensional distribution r E S'(10, oo)) [6] Work supported by the Bundesministerium fiir Forschung und Technologie. 360
(T, ~o) = (r, ~),
~0E S(R4).
(2)
The function ~ E S(10, oo)) is defined by:
~(s) = f dxe(Xo)8(x2-s)~(x).
(3)
R4
The Fourier transform 7~ of T can be written in the same way. The positive definiteness property forces the corresponding one-dimensional distribution to be a positive measure/a: oo
(T, ~0) = f d~t(s)~(s), 0
(4)
The distribution r o f formula (2) can be written as the distributional derivative of a polynomially bounded, continuous function g E C°(] 0, '~)), i.e.,
(r, ~) = J dsg(s)-~(l)(s) .
(5)
0 By definition lims~o÷S~g(s) exists for/3 ~> 0, but it may happen that this limit exists for some/3 < 0, too [7]. If/3 o is the infimum of all these values/3 then we call l +/3 o the degree of singularity of T on the light cone (SG [T]). The meaning of SG is illustrated by the following integral representation for T, which is a consequence of relations (2) and (5): eo
T(x)
=
fdsg(s)e(Xo)Sq)(x2-s).
(6)
0 The reader may convince himself that SG does not depend on the special choice of l and g in formula (5). The distribution e(Xo)8(l)(x2) for instance has SG = l,
Volume 52, number 3
PHYSICS LETTERS
whereas SG = _co for any distribution T if its support does not contain the light cone x 2 = 0. Theorem: Let T a n d T b e as in eqs. (6) and (4). Then: a) i f a > SG [T] - 1 then f~dtl(s)s-~< oo; b) if f~o d/a(s)s_ a < oo for s > 0 then SG [T] ~< 1 + s. Thus, if % : = inf {s/> 0; f~°dla(s)s-~<°°} then SG [T] = s o + 1 •
(7)
14 October 1974
As Xk is continuous and bounded for s ~> 1 the integral is convergent because e > 0 can be chosen arbitrarily small, so that l +/3 - 2 - a = SG [T] - 2 a + e < - 1 . This proves part a). Proof of part b): Starting with eq. (4) for the Fourier transform Twe get for T the spectral-representation [9]:
Note that this statement on the asymptotic behaviour of the measure/2 does not depend on the singularity structure in the time like region x 2 > 0. The singularities in that region are restricted by the positivity o f the measure/a [8] as one can see explicitly in simple examples of the type e(Xo)8 (x 2) + be(xo)6 (/)(x2-s). The Fourier transform of this expression is positive only if b = 0. Proof o f part a): We start with a representation (6) for T. By definition of SG [T] the polynomially bounded function g E C°([ 0, ~)) satisfies lims~o+S#g(s) = 0 with fl = SG [T] - l + e for any e > 0. Computing the Fourier transform o f relation (6) one gets the connection between the measure/2 and the function g:
f(s') = f dla(s)s-l/2jl(x/~') 0
f du(s)~(s) =f dssk+i+e-1x~(sl~k+l)(s).
is then well defined and continuous for s' > 0. Using the bound:
o
(8)
o
ao
Xk(S) = 2 k + l - l f au u -f3-(k+l)/2 0
Because of the polynomial bound
[g(u/s)(u/s)~l <~C(1 + u/s) n ,
s > O,
it is easy to prove (using Lebesque's theorem on dominated convergence) that for sufficiently large k the function Xk is continuous for s > 0 and vanishes for s -+ co. Equality (8) can be extended to the function ~(s) = O(s- 1)s --~ with s < SG [7] - 1. Eq. (8) then reads
=s ..
Applying a test function e E S(R 4) to the 'Pauli-Jordan' function A(x, s) gives
( A(x, s), ~o)= - f ds's't(x/~')-tJt(x/~')¢ (t+l )(s') o (lO) Here l may be any non negative integer with The function f
l/2>/a.
o0
y t> 0
one can apply Lebesgue's theorem on dominated convergence to see that lims_. 0 +s'ef(s ') exists for 0 < e: = 1/2 - a < l/2. If one combines eqs. (9) and (10) one has a representation of type (6) for T 0o
X J_k_l(4ff)g(u/s)(u/s)~.
1
(9)
0
IJ/(y)l ~< ClYl(1 + y l ) - I ,
Here k is an arbitrary integer, sufficiently large so that the following integral converges (Jk = Bessel function):
f
T(x) = f dp(s)A(x, s).
f a, ,'+,-= -ox,,(s).
r(x) = ( - 1 ) l+1 Jds's'l/2f(s')e(Xo)6l+l(x2-s). 0 The function s'l/2f(s ') plays the r91e o f g and l is replaced by l + 1. As lim s,+0+(s ') -I/2-es'll2f(s') exists, one has by definition of SG: SG[T]
<-t/2-e(l+l)=a+
l.
This proves part b). We come back now to e+-e - annihilation into hadrons. If we assume one-photon exchange then the total cross section can be written as Otot(S) = 16rr 3" S 20(s)/s,
S = (Pe -+Pe +)2
1
361
Volume 52, number 3
PHYSICS LETTERS
where p(s) is just the spectral function o f the measure/a:
r(x) =
f
References
Our theorem shows that in all field theoretical models the leading asymptotic behaviour of Otot(S) is determined by the light cone structure of (01 []u(x),]~(0)] 10). Because of positivity the contributions of the singularities in the time like region are less important [8]. Assume for example that Otot(S) behaves asymptotically, up to logarithmic corrections, as some power S b , i.e.,
lim s-b+e otot(S)
(O
for
e
for
e> 0
According to formula (7) the distribution T in eq. (1) then has SG [T] = 3 + b. Otot(S) ~ const. (powers of logs) requires SG [T] --- 3. The behaviour Otot ~ s -1 (powers of log s) means SG [T] = 2. This is of course the well known case of asymptotically free theories.
362
I would like to thank Professor H.A. Kastrup for suggesting the problem and for a critical reading of the manuscript.
,is p(s)~(x, s).
(2m~r)2
s~ ~
14 October 1974
[11 R. Brandt, Phys. Rev. D1 (1970) 2808. [2] B. Richter, invited talk in: Proc. Conf. on Lepton induced reactions, Irvine California, Dec. 1973, to be published. [3] M. Magg, contribution no. 149 in: Proc. Symp. on Electron and photon interactions at high energies, Bonn, August 1973; and Rigorous results concerning cone dominance in deep inelastic lepton-hadron scattering, RWTH Aachen preprint, Jan. 1974, [4] R. Gatto, Summary report at the lI-Aix-en-Provenee Intern. Conf. on Elementary particles, September 1973. [5] R.F. Streater and A.S. Wightman, PCT, spin and statistics and all that (Benjamin, New York, 1964). [61 P. Meth~e, Comm. Math. Helv. 28 (1954) 225. [7] O. Steinmann, J. Math. Phys. 4 (1963) 583. [8] L. Schwartz, Th6orie des distributions (Hermann, Paris, 1966). [9] G. Kgllen, HeN. Phys. Acta 25 (1952) 416.
PHYSICS LETTERS
LIGHT CONE STRUCTURE AND ASYMPTOTIC
14 October 1974
OF THE CURRENT BEHAVIOUR
TWO-POINT FUNCTION
IN MOMENTUM
S P A C E ¢~
M. MAGG
Institut fiir TheoretischePhysik, TechnischeHochschuleAachen, FR Germany Received 19 September 1974 We prove the correspondence between the degree of singularity of a two point function on the light cone and the asymptotic behaviour of its Fourier transform. The result is relevant for e+-e - annihilation. Contrary to deep inelastic e-p scattering [e.g., 1 ] recent experiments [2] seem to indicate that e+-e annihilation requires a stronger singular structure than the canonical one for the current commutator. For this interpretation it is crucial that in both processes a) one-photon exchange dominates b) asymptotic behaviour in m o m e n t u m space can be uniquely translated into space-time singularities of the current commutator on the light cone. In the following we shall assume the validity of point a). Regarding the Bjorken limit in e-p scattering a complete discussion of point b) is given in ref. [3]. As the preliminary experiments with e+-e - annihilation seem to contradict many fashionable theoretical models [4] it may be worth-while to investigate the relation between the current two-point function in configuration space and the asymptotic behaviour of otot(s) independent of any special model, using only general principles of quantum field theory like Lorentz-invariance, locality and positivity [5]. The total cross section for e+-e - annihilation into hadrons is proportional to the Fourier transform of the following distribution T:
(O[(ju(x),iv(Y)] lO) =(~xx~ ~x -guvff])T(x-y)"
(1)
T is a tempered, Lorentz-invariant, uneven distribution on the configuration space R 4. It is the antisymmetric part of the positive definite Wightman function (0 Iju(x)ju(y)l 0). Any distribution of this type can be represented by a uniquely determined one dimensional distribution r E S'(10, oo)) [6] Work supported by the Bundesministerium fiir Forschung und Technologie. 360
(T, ~o) = (r, ~),
~0E S(R4).
(2)
The function ~ E S(10, oo)) is defined by:
~(s) = f dxe(Xo)8(x2-s)~(x).
(3)
R4
The Fourier transform 7~ of T can be written in the same way. The positive definiteness property forces the corresponding one-dimensional distribution to be a positive measure/a: oo
(T, ~0) = f d~t(s)~(s), 0
(4)
The distribution r o f formula (2) can be written as the distributional derivative of a polynomially bounded, continuous function g E C°(] 0, '~)), i.e.,
(r, ~) = J dsg(s)-~(l)(s) .
(5)
0 By definition lims~o÷S~g(s) exists for/3 ~> 0, but it may happen that this limit exists for some/3 < 0, too [7]. If/3 o is the infimum of all these values/3 then we call l +/3 o the degree of singularity of T on the light cone (SG [T]). The meaning of SG is illustrated by the following integral representation for T, which is a consequence of relations (2) and (5): eo
T(x)
=
fdsg(s)e(Xo)Sq)(x2-s).
(6)
0 The reader may convince himself that SG does not depend on the special choice of l and g in formula (5). The distribution e(Xo)8(l)(x2) for instance has SG = l,
Volume 52, number 3
PHYSICS LETTERS
whereas SG = _co for any distribution T if its support does not contain the light cone x 2 = 0. Theorem: Let T a n d T b e as in eqs. (6) and (4). Then: a) i f a > SG [T] - 1 then f~dtl(s)s-~< oo; b) if f~o d/a(s)s_ a < oo for s > 0 then SG [T] ~< 1 + s. Thus, if % : = inf {s/> 0; f~°dla(s)s-~<°°} then SG [T] = s o + 1 •
(7)
14 October 1974
As Xk is continuous and bounded for s ~> 1 the integral is convergent because e > 0 can be chosen arbitrarily small, so that l +/3 - 2 - a = SG [T] - 2 a + e < - 1 . This proves part a). Proof of part b): Starting with eq. (4) for the Fourier transform Twe get for T the spectral-representation [9]:
Note that this statement on the asymptotic behaviour of the measure/2 does not depend on the singularity structure in the time like region x 2 > 0. The singularities in that region are restricted by the positivity o f the measure/a [8] as one can see explicitly in simple examples of the type e(Xo)8 (x 2) + be(xo)6 (/)(x2-s). The Fourier transform of this expression is positive only if b = 0. Proof o f part a): We start with a representation (6) for T. By definition of SG [T] the polynomially bounded function g E C°([ 0, ~)) satisfies lims~o+S#g(s) = 0 with fl = SG [T] - l + e for any e > 0. Computing the Fourier transform o f relation (6) one gets the connection between the measure/2 and the function g:
f(s') = f dla(s)s-l/2jl(x/~') 0
f du(s)~(s) =f dssk+i+e-1x~(sl~k+l)(s).
is then well defined and continuous for s' > 0. Using the bound:
o
(8)
o
ao
Xk(S) = 2 k + l - l f au u -f3-(k+l)/2 0
Because of the polynomial bound
[g(u/s)(u/s)~l <~C(1 + u/s) n ,
s > O,
it is easy to prove (using Lebesque's theorem on dominated convergence) that for sufficiently large k the function Xk is continuous for s > 0 and vanishes for s -+ co. Equality (8) can be extended to the function ~(s) = O(s- 1)s --~ with s < SG [7] - 1. Eq. (8) then reads
=s ..
Applying a test function e E S(R 4) to the 'Pauli-Jordan' function A(x, s) gives
( A(x, s), ~o)= - f ds's't(x/~')-tJt(x/~')¢ (t+l )(s') o (lO) Here l may be any non negative integer with The function f
l/2>/a.
o0
y t> 0
one can apply Lebesgue's theorem on dominated convergence to see that lims_. 0 +s'ef(s ') exists for 0 < e: = 1/2 - a < l/2. If one combines eqs. (9) and (10) one has a representation of type (6) for T 0o
X J_k_l(4ff)g(u/s)(u/s)~.
1
(9)
0
IJ/(y)l ~< ClYl(1 + y l ) - I ,
Here k is an arbitrary integer, sufficiently large so that the following integral converges (Jk = Bessel function):
f
T(x) = f dp(s)A(x, s).
f a, ,'+,-= -ox,,(s).
r(x) = ( - 1 ) l+1 Jds's'l/2f(s')e(Xo)6l+l(x2-s). 0 The function s'l/2f(s ') plays the r91e o f g and l is replaced by l + 1. As lim s,+0+(s ') -I/2-es'll2f(s') exists, one has by definition of SG: SG[T]
<-t/2-e(l+l)=a+
l.
This proves part b). We come back now to e+-e - annihilation into hadrons. If we assume one-photon exchange then the total cross section can be written as Otot(S) = 16rr 3" S 20(s)/s,
S = (Pe -+Pe +)2
1
361
Volume 52, number 3
PHYSICS LETTERS
where p(s) is just the spectral function o f the measure/a:
r(x) =
f
References
Our theorem shows that in all field theoretical models the leading asymptotic behaviour of Otot(S) is determined by the light cone structure of (01 []u(x),]~(0)] 10). Because of positivity the contributions of the singularities in the time like region are less important [8]. Assume for example that Otot(S) behaves asymptotically, up to logarithmic corrections, as some power S b , i.e.,
lim s-b+e otot(S)
(O
for
e
for
e> 0
According to formula (7) the distribution T in eq. (1) then has SG [T] = 3 + b. Otot(S) ~ const. (powers of logs) requires SG [T] --- 3. The behaviour Otot ~ s -1 (powers of log s) means SG [T] = 2. This is of course the well known case of asymptotically free theories.
362
I would like to thank Professor H.A. Kastrup for suggesting the problem and for a critical reading of the manuscript.
,is p(s)~(x, s).
(2m~r)2
s~ ~
14 October 1974
[11 R. Brandt, Phys. Rev. D1 (1970) 2808. [2] B. Richter, invited talk in: Proc. Conf. on Lepton induced reactions, Irvine California, Dec. 1973, to be published. [3] M. Magg, contribution no. 149 in: Proc. Symp. on Electron and photon interactions at high energies, Bonn, August 1973; and Rigorous results concerning cone dominance in deep inelastic lepton-hadron scattering, RWTH Aachen preprint, Jan. 1974, [4] R. Gatto, Summary report at the lI-Aix-en-Provenee Intern. Conf. on Elementary particles, September 1973. [5] R.F. Streater and A.S. Wightman, PCT, spin and statistics and all that (Benjamin, New York, 1964). [61 P. Meth~e, Comm. Math. Helv. 28 (1954) 225. [7] O. Steinmann, J. Math. Phys. 4 (1963) 583. [8] L. Schwartz, Th6orie des distributions (Hermann, Paris, 1966). [9] G. Kgllen, HeN. Phys. Acta 25 (1952) 416.