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Crossing sum rules and high-energy bounds
Nuclear Physics B53 (1973) 125- 134 North-Holland Pubhshmg Company
CROSSING SUM RULES AND HIGH-ENERGY BOUNDS P. GRASSBERGER Physikalisches Instltut der Umr'ersztat Bonn
H. KtJHNELT * instltut fur Theoretische Phystk der Untversitdt Karlsruhe
Received 6 November 1972 Abstract: Exact energy-averaged high-energy bounds for total ~rn cross sections are gwen which do not depend on the d-wave scattering lengthes. Instead, the low-energy input is, essentially, the properties of the fo meson. The starting points are rigorous sum rules for the physical absorptive parts, following from crossing and analyticity Except at rather low energies, these bounds are much ttghter than the best prevtous ones
1. Introduction Some years ago, Yndurain and Common [ 1] have shown that one can use crossing symmetry to establish rigorous connections between low- and high-energy scattering. For n ° n ° scattering, they found that knowledge of the d-wave scattering length allows one to get absolute bounds on the energy-averaged total cross section. For infinite energy, they correspond to the Froissart bound with fixed constants. Subsequently, Steven [2], Blankenbecler and Savlt [3], and Roy [4] have used optimization methods (see, e.g., ref. [5]) to get the best bounds obtainable from knowledge of the scattering lengths. For rather low energies (Ec.m. ~ 1 GeV, say), these bounds are of the order of 3 0 - 7 0 mb but they deteriorate quickly at higher energies. Additional ad hoc assumptions (incluchng constant total cross sections) reduced them to ~ 40 mb at high energy. The scattering lengths used in these bounds are of course not directly measurable, and their most reliable estimates stem from fixed-t dispersion relations [6]. Now, the starting point for the high-energy bounds are also the fixed-t dispersion relations (or, equivalently, Frotssart-Gribov representations). So, one might wonder whether one has been trapped in some vicious circle, regaining just the physical information that one has put in before. The reason why one is not trapped, of course, is that both * Alexander yon Humboldt Fellow. Present address lnstltut fur Theoretlsche Physlk, Umversltat W~en.
P. (;rassberger,1t. Ki~hnelt,Crossingsumrules
126
times one uses dispersion relations in different channels *. The two dispersion relations for the d-wave scattering length m n°Tr° scattering are indeed ** a00 - 157r8 4f x '~
A 00(x,4) ,
(1.1)
and [71
8 f4 dXlAOO(x,O)÷4x(x-2) OAOO(x,t=O))
(1.2)
The above argument shows, however, that one really uses just the equation
/
A1 = / d X x3
t
AOO(x,4)__AOO(x,O) 4x(X-(x4) 22) ~t ~ AOO(x't=O) = 0
,
(1.3)
which ~s a relatmn between physical absorptive parts only. It suggests that a more straightforward approach should start from eq. (1.3) directly. Indeed, this will be done in the present paper. The practical gain compared to the results of Steven [2] is especially important at higher energies. There, it deteriorates at a much slower rate. In some way, our results can be compared with those of Blankenbecler and Savit [3], who started from eq. (1.1), but took as input the scattering length and some low-energy phases. The main practical advantage of our method consists in avoiding spurious errors resulting from treating the poorly known s-waves differently in eq. (1.1) and eq. (1.2) the lowest angular momentum contributing to eq. (1.3) is I = 2, and the only input needed is the low-energy d-wave (i.e. the fo meson). Even w~th moderate estimates on its accuracy, we expect our bounds to have errors of less than l lYJ6. For n+Tr° scattering, the situation is similar, though not quite as simple. Using essentially the same input, we get bounds of about the same magnitude as for that is Otot ~ 90 mb for s ~ 10 GeV 2.
n°n°,
2. Neutral pions Let us first give an alternative proof of the crossing sum rule eq. (1.3). Such sum rules (or physical region crossing relattons) have first been discussed by Wanders and Roskies [8, 9]. We use here the formulation of ref. [10], m which they are generally written as * One should, however, keep this m mind when comparing the bounds at too low energies with experimental data, as the two dispersion relanons have very similar low-energy input. Only their high-energy contributions are substantially different. ** We use units m which/t = c = m n = 1; the scattering length is defined as
=
k- 2,-1 k~0
),
a00
* :4)
P. Grassberger, H. Kuhnelt, Crossing sum rules
127
3
f dx X(x;t,, 9' tk)AOO(x't,) = 0 . i=1
(2.1)
4
I,k cyclic Here, 0 ~< t 1 , t2, t 3 < 4, and
( t / - tk)(Zx - 4 + ti) X(x; t,, t/, tk) = (x -. tl)(x - tk)(X-- 4 + t i + t/)(x -- 4 + t + tk) "
(2.2)
Eq. (1.3) is simply got from eq. (2.1) by first differentiating it with respect to tl, and taking then the limits * tl = t 2 - - * 0 ,
t3-+4.
Inserting into it the (convergent) partial wave decompositions ofA O0(x,t),
AOO(x't)= ~ x - ~ 4 ~ ( 2 ' + l ) P l ( l + x 2 ~ - t 4 ) a O O ( x ) '
(2.3)
1=0 we can write eq. (1.3) as ~ ( 2 l + 1) dx al(x)aOO(x)= O, l=2 4 I even with
1
\(PI~,~-4{x+ 4 ,]~-
(2.4)
1
x(x - 2)4l(1 + 1) )
(2.5)
First, notice that the s-wave does not appear in eq. (2.4): it starts with the d-wave. Secondly, we notice that (for x > 4)
~l(X) > O,
for
ec2(x) < 0 .
1 ~> 4 ,
(2.6) (2.7)
Finally, otl(x ) is monotonically increasing with l (for fixed x > 4), which will allow us to apply Martins [ 11 ] solution of the following maximum problem. Choosing three energy values s 2 > s I i> s O > 4, we get, on account of eqs. (2.6) and (2.7), and because of the unitarity constraints
o < oo(x)
I,
(2.8)
the inequality
* We should mention that interesting mequahties on low-energy absorptive parts follow from the hmit t3 ~ O, after taking the appropriate number of derivatives [8, 9] (see also sect. 4).
P. Grassberger, 1t. Kuhnelt, Crossing sum rules
128
s2
(2t+ i) f /=2
sl
<~-5 f
o dx a2(x)aOO(x) I! - 5
4
+
dx a2(x) - C 1 .
(2.9)
s2
Our problem is now to find the maximal value allowed for an averaged total cross section, compatible with the constraints (2.8) and (2.9). The averaged total cross section is defined by $2
*rO*rO f f tot
---sl
*tono dxq(x) Otot (X)
= 32n G (21+ 1) f dx I=0 st
aOO(x),
(2 10)
with some normalized weight function q(x) to be determined later. The solution of this kind of problem is well known [ 11, 2, 4]. The maximum of -- ffOffO O tot IS assumed if the aOO(x) are given by 1,
a°°(x)=
for
0,
for
l<~L(x), t>L(x),
(2.11)
where L(x) is defined by x -4 x 4 OtL(x)(X) ~ ~ K "~ OtL(x) + 2(x) q ( x ) ' S2 f
L(x) dx~ ( 2 / + l)al(x)=-C I .
sl
(2.12)
(2.13)
1=2
Here, K ts some constant independent ofx. It is determined by eqs. (2.12) and (2.13). Numerically, the evaluation of the bound Is straight-forward. We used weight functions
q(x) -
const X6
with 0 ~< 8 ~< 2. The result was not very much dependent on the particular choice, and finally we settled on _ _
1
q(x) - x In (s2[sl)
(2.14)
P. Grassberger, tt. Kithnelt, Crossing sum rules
129
For the low-energy isoscalar d-wave, we used a modified Breit-Wlgner ansatz for the fo meson, which reproduced the correct scattering length [6] as well: F(s)2m 2
al2=O(s)=(mff _ s) 2 +
(2.15)
r(s)2m2'
with F(s)
2k
rf t(m - 4)(s + a)
a = 68.0,
~=
156 MeV.
(m 2 _ 4)-~ '
For the I = 2 d-wave, we used
a/=2(s) = k2 (k2(a + 4))4a/= 2 \
(2.16)
s+a
With these ansatz, we got C 1 ~ 1.06 × 10 - 6 (with s o = min (Sl, 200 m 2) , s I t> 120 m2). The resulting high-energy bounds, for s 2 = 10 Sl, are shown in fig. 1. For other ratios o f s 2 / s l , the bounds are shghtly higher. It is interesting that the bounds are only very weakly dependent on the low-energy input C 1. Thus, a change of C 1 by 30% always resulted in less than 10°6 change in the bounds. This is easily understood by the rapid increase of otI with increasing 1. It shows that improvements should not so much be searched by using more accurate low-energy input but by using some additional kind of information either on the lowor on the high-energy phases.
mb
j/ // /,11"
j~ t ~
50-
• S
Fig. 1. Upper bounds to energy-averaged norro total cross sections. The horizontal bars indicate the averagmg mtervals.
P Grassberger,tl. Kuhnelt, Crossingsum rules
130
Finally we should mention that asymptotically, our bound becomes equivalent to the Frolssart bound, o-- 71.O,/1.O tot ~ 7r( In s) 2. (The factor n in front cannot be improved without additional high-energy assumptions, in contrast to speculations in ref. [3]). Looking at fig. 1, one would suggest fitting a parabola o-~ono tot ~ const × (In s) 2, with const ~ 0.5 '< 7r. This shows very clearly that the curve shown m fig. I is not yet the parabola corresponding to the Froissart bound *
3. C h a r g e d
pions
For charged pions, one has two additional independent crossing sum rules [12]. Only one of them will be used here, tt is A2(tl' t2)--
×
f (X -
4
dx
tl)(X - t2)(X - 4 + t 1 + t2)
x44 Al(x, tl
AO(x, tt)-5A2(X, t l ) - 3
x + 2t 1 -
/
-[tl~t21
=0
(3.1)
Here, A I are the absorptive parts in the lsospm = I channels, and 0 ~< t 1, t 2 ~< 4. Since we want to constrain o ~r+'r° = ½(o 1 + o2), we need a sum rule getting high-energy contributions mainly from I :# 0. The one used in the following is (for A 1, see eq. (1.3)) 6A 1 - A2(4,0 ) + 4 ~ A2(t = 0,0) = ~ (21+ 1 ) ~ 1=2 1=0
l+l even
fdx~(x)a~(x)=O, 4
(3.2)
with 8
[p (x+4
'~
~l(x)-
x~(x_4){ L i ~ ] -
~(x)=3
3 x - 4 ~ I -x -~ 4x~(x- 4)2 3. . . .
/32(x)_ T--9 x - 1 6 --3
xT(x - 4)'i
21(l+ l ) x ] 1 + ~x ~ 4 ~ '
(x+4~
P / \ ~ - ]
[pl (kx~+_ 44 ~] - 1 1
-
1 -
x4_~(/2
(3.3)
+ 1-
2)
]
,
41(1+ 1)_ ( 9 x - 28). x{(x - 4)r
(3.4)
(3.5)
This combination is chosen because of the following properties of the ~3/(x): * After completion of this work, we found that similar results have recently be obtained by F. Ynduram, CERN-TH. 1554 (1972).
P. Grassberger, 1t. Kiihnelt, Crossing sum rules
(i) #t°(x) < O,
/3)(x)> O,
#2(x) > O,
131
(3.6)
for a l l l ~ > 2 , x>4; (i0 - ~ ( x ) , #}(x) and #~(x) are monotonically increasing with I (at fixed x > 4); 0ii) for l and x going simultaneously to infinity, m such a way that {x+4'~
et \ x : -~ / ~>O(x), we have
(x+4
~¢(x) ~ . ~ ( x ) ~ x -3 P l \ x _ 4 ,1'
(3.7)
while
#to(X) ~ X -4 el k x - - 4 - , 1 "
(3.8)
Since m addition, for fixed l, #~(x) ~ x -5 ,
(3.9)
we see that high-energy and large-/isoscalar contributions to the sum rule are suppressed. Indeed, one can use inequalities resulting from the crossing sum rule eq. (2.1) (ref. [9]; see also sect. 4) to show that already the contribution of the I = 0, l = 4 wave should be less than -~.of the contribution of the I = 0 d-wave. The higher I = 0 partial waves are completely negligible. We use eq. (3.2) by rewriting it as $2
~(21+ l=2 ~<
1) f dxl~](.;c)a](x)+~2(x)al2(x)] sl
dxll~to(x)laO(x)-C2 ,
( 2 l + 1) l=2
(3.10)
4
and use C2, the contribution of the isoscalar absorptive parts, as input. Because of the above-mentioned properties of #°(x), we can write (3.11) C2 < 7~X 5 f d x I~(x)la°(x) ~. 2.2 X 10 -5 4 (The situation concerning the higher isoscalar partial waves could partially be further improved by taking more refined combinations of the sum rules (2.1), (3.1) and a third sum rule not stated here [ 12]). Here we have again used the parametrization eq. (2.15). The further treatment parallels exactly the one for neutral p~ons, with the only exception that we now calculate bounds on o-'r+"° tot • The results are shown in fig. 2.
132
P. Grassberger, tl. Kia'hnelt, Crossing sum rules mb
'~tm
150-
/
/
/
/,d ...=e,./=(" _~/ic" =I=
50-
I:lg. 2. Upper bounds to n+n o total cross secnons. The bound is nearly the same as for neutral plons, except for rather low energies (s ~ 100 - 500 m2), where it is shghtly better.
4. Discussion and further improvements We hope to have clarified somewhat the role played by crossing symmetry, and especially the crossing sum rules, in deriving high-energy bounds. While the improvement over previous treatments [2, 4] was not very large in the low-energy domain, it was quite remarkable at higher energies. A direct comparison with the results of refs. [2, 4] is difficult since these authors always averaged over regions going down to threshold (i.e., they took s I --- 4). Furthermore, Roy [4] uses a weight function peaked strongly at low energies. For better comparison, we have redone their 7r°1r° calculations with our weight function (2.14) and s I = 2.4 GeV 2, s 2 = 24 GeV 2. We used a~ 0= 0.00073, and found a bound about 3 times as large as our hound. The results of Blankenbecler and Savit [3] are on the one hand less easy to compare since they used some extra assumptions on the high-energy cross sections. On the other hand, their method is similar to ours in taking into account also some lowenergy data from the physical region. Our approach seems, however, to be cleaner than theirs, which depends crucially on concellations in the low-energy input. These are automatically taken into account in the present work. For instance, our bounds are completely independent of the low-energy s- and p-waves. Improvements of our bounds could be made by either supposing some additional high-energy properties, or by changing the low-energy input. In the high-energy domain, one can assume e.g. the diffraction peak width, the ratio of total to elastic scattering or a constant cross section. What improvements can can be expected can be traced, e.g. from the review of Roy [4].
P. Grassberger, tL Kiihnelt, Crossing sum rules
133
On the low-energy side, one could assume the I -- 4-wave in addition to the dwave, and subtract its c o n t r i b u t i o n . However, this implies strong concellations between poorly k n o w n experimental quantities, and is subject to a similar criuque as ref. [3]. A cleaner, though less powerful, way is to use more i n f o r m a t i o n from the crossing sum rules. Let us sketch what one can do with neutral pions. Differentiating eq. (2.1) twice with respect to t l , and once with respect to t2, and taking t t = t 2 = t 3 ~ 0 at the end, one gets
dxTl(x)aOO(x) = O,
(2•+ 1) l=2
(4.1)
4
with
(x
x - (x - 4) { (x One sees that 7/(x) has similar properties as al(x ) 7l(x) > 0 ,
for
I/> 4 ,
72(x ) < 0 .
Indeed, the essential difference with respect to the a I is that n o w the Pl(z) are replaced by their Taylor expansions up to second order. As a consequence, eq. (4.1) does n o t lead to the Froissart b o u n d , b u t allows Oto t to increase like s~ -e, e > 0. For finite energies, however, using eq. (4.1) in the same way as eq. (2.4) above, we get results very similar to fig. 1 * A more complete m e t h o d would consist in treating eqs. (4.1) and (2.4) as simultaneous, i n d e p e n d e n t constraints. Finally, one can show that
~(x)-= st(x) .... vt(x),
•
has the following properties, a'2(x) < 0, ~/(x) > 0 for 1/> 4, and ~/(x) is m o n o t o n ically increasing with l. So, one can use also * For completeness, we should mention that these bounds are very similar to the bounds on resonance couphngs found In ref. [9]. For charged pions, we can shghtly improve a bound on the d- and f-waves [91 by using eq. (4.1) and a similar sum rule obtained from A2(tl, t2) (see eq. (3.1)) Subtracting these two from each other, we get
i x x,x 4,, , {.o,x>
6
Assuming here narrow resonance saturation, we see that
o >6(mf° 9 \mg /
} >o
134
P. Grassberger, tL Kithnelt, Crossing sum rules
(21 + 1) dx ~l(x)aOO(x) = 0 , (4.3) 1=2 4 as a starting point, instead ofeq. (2.4). Since on the one hand ~'2(m2o) "~ o~2(m2o), but ~/(x) ~- og(x ) for the values of/dominating the averaged cross sections at s 1, s2 -. o% one expects the bounds following from eq. (4.3) to be asymptotically better than the bound following from eq. (2.4). We have looked at this numerically. For the d-wave parametnzation gwen in sect. 2, we found that using eq. (4.3) instead of (2.4) indeed improves the bounds for s 1 ~ 400 GeV 2. For lower energies, this new bound Is however slightly worse: its lowest value, at s ~ 5 GeV 2, is 130 rob. We t h a n k Dr. D. Schwela for p o i n t i n g o u t t w o errors in the o n g m a l m a n u s c r i p t .
References [1] F.J. Ynduram, Phys Letters 31B (1970) 368, A.K. Common, Nuovo ('lmento A69 (1970) 115; A.K. Common and F.J. Ynduram, Nucl Phys. B26 (1971) 167. 12] A C. Steven, Cambrtdge preprmt IlEP 72--9 (1972). 131 R. Blankenbecler and R. Savlt, Phys. Rev. D5 (1972) 2757. [41 S.M. Roy, Physics Reports 5C (1972) 125. 15] M. Einhorn and R. Blankenbecler, Ann. of Phys. 67 (1971) 480; R. Eden, Lectures at Kalserlautern Summer School (1972) 16] D. Morgan and J. Pis6t, Springer tracts m modern physics, vol.55 (1970). 17] S. Krlnsky, Phys. Rev. 1)4 (1971) 1096. [8] G. Wanders, Nuovo Cimento 63A (1969) 108; GIFT-Seminar, 1971. [9] R. Rosktes, Phys Rev. 1)2 (1970) 247, 1649. [101 P Grassberger, Nucl. Phys. B42 (1972)461, P. Grassberger and I1. Kuhnelt, Nuovo Cimento Letters 4 (1972) 895 [11] A. Martin, Phys. Rev. 129 (1963) 1432 [12] P. Grassbcrger and H. Kuhnelt, to be pubhshed.
CROSSING SUM RULES AND HIGH-ENERGY BOUNDS P. GRASSBERGER Physikalisches Instltut der Umr'ersztat Bonn
H. KtJHNELT * instltut fur Theoretische Phystk der Untversitdt Karlsruhe
Received 6 November 1972 Abstract: Exact energy-averaged high-energy bounds for total ~rn cross sections are gwen which do not depend on the d-wave scattering lengthes. Instead, the low-energy input is, essentially, the properties of the fo meson. The starting points are rigorous sum rules for the physical absorptive parts, following from crossing and analyticity Except at rather low energies, these bounds are much ttghter than the best prevtous ones
1. Introduction Some years ago, Yndurain and Common [ 1] have shown that one can use crossing symmetry to establish rigorous connections between low- and high-energy scattering. For n ° n ° scattering, they found that knowledge of the d-wave scattering length allows one to get absolute bounds on the energy-averaged total cross section. For infinite energy, they correspond to the Froissart bound with fixed constants. Subsequently, Steven [2], Blankenbecler and Savlt [3], and Roy [4] have used optimization methods (see, e.g., ref. [5]) to get the best bounds obtainable from knowledge of the scattering lengths. For rather low energies (Ec.m. ~ 1 GeV, say), these bounds are of the order of 3 0 - 7 0 mb but they deteriorate quickly at higher energies. Additional ad hoc assumptions (incluchng constant total cross sections) reduced them to ~ 40 mb at high energy. The scattering lengths used in these bounds are of course not directly measurable, and their most reliable estimates stem from fixed-t dispersion relations [6]. Now, the starting point for the high-energy bounds are also the fixed-t dispersion relations (or, equivalently, Frotssart-Gribov representations). So, one might wonder whether one has been trapped in some vicious circle, regaining just the physical information that one has put in before. The reason why one is not trapped, of course, is that both * Alexander yon Humboldt Fellow. Present address lnstltut fur Theoretlsche Physlk, Umversltat W~en.
P. (;rassberger,1t. Ki~hnelt,Crossingsumrules
126
times one uses dispersion relations in different channels *. The two dispersion relations for the d-wave scattering length m n°Tr° scattering are indeed ** a00 - 157r8 4f x '~
A 00(x,4) ,
(1.1)
and [71
8 f4 dXlAOO(x,O)÷4x(x-2) OAOO(x,t=O))
(1.2)
The above argument shows, however, that one really uses just the equation
/
A1 = / d X x3
t
AOO(x,4)__AOO(x,O) 4x(X-(x4) 22) ~t ~ AOO(x't=O) = 0
,
(1.3)
which ~s a relatmn between physical absorptive parts only. It suggests that a more straightforward approach should start from eq. (1.3) directly. Indeed, this will be done in the present paper. The practical gain compared to the results of Steven [2] is especially important at higher energies. There, it deteriorates at a much slower rate. In some way, our results can be compared with those of Blankenbecler and Savit [3], who started from eq. (1.1), but took as input the scattering length and some low-energy phases. The main practical advantage of our method consists in avoiding spurious errors resulting from treating the poorly known s-waves differently in eq. (1.1) and eq. (1.2) the lowest angular momentum contributing to eq. (1.3) is I = 2, and the only input needed is the low-energy d-wave (i.e. the fo meson). Even w~th moderate estimates on its accuracy, we expect our bounds to have errors of less than l lYJ6. For n+Tr° scattering, the situation is similar, though not quite as simple. Using essentially the same input, we get bounds of about the same magnitude as for that is Otot ~ 90 mb for s ~ 10 GeV 2.
n°n°,
2. Neutral pions Let us first give an alternative proof of the crossing sum rule eq. (1.3). Such sum rules (or physical region crossing relattons) have first been discussed by Wanders and Roskies [8, 9]. We use here the formulation of ref. [10], m which they are generally written as * One should, however, keep this m mind when comparing the bounds at too low energies with experimental data, as the two dispersion relanons have very similar low-energy input. Only their high-energy contributions are substantially different. ** We use units m which/t = c = m n = 1; the scattering length is defined as
=
k- 2,-1 k~0
),
a00
* :4)
P. Grassberger, H. Kuhnelt, Crossing sum rules
127
3
f dx X(x;t,, 9' tk)AOO(x't,) = 0 . i=1
(2.1)
4
I,k cyclic Here, 0 ~< t 1 , t2, t 3 < 4, and
( t / - tk)(Zx - 4 + ti) X(x; t,, t/, tk) = (x -. tl)(x - tk)(X-- 4 + t i + t/)(x -- 4 + t + tk) "
(2.2)
Eq. (1.3) is simply got from eq. (2.1) by first differentiating it with respect to tl, and taking then the limits * tl = t 2 - - * 0 ,
t3-+4.
Inserting into it the (convergent) partial wave decompositions ofA O0(x,t),
AOO(x't)= ~ x - ~ 4 ~ ( 2 ' + l ) P l ( l + x 2 ~ - t 4 ) a O O ( x ) '
(2.3)
1=0 we can write eq. (1.3) as ~ ( 2 l + 1) dx al(x)aOO(x)= O, l=2 4 I even with
1
\(PI~,~-4{x+ 4 ,]~-
(2.4)
1
x(x - 2)4l(1 + 1) )
(2.5)
First, notice that the s-wave does not appear in eq. (2.4): it starts with the d-wave. Secondly, we notice that (for x > 4)
~l(X) > O,
for
ec2(x) < 0 .
1 ~> 4 ,
(2.6) (2.7)
Finally, otl(x ) is monotonically increasing with l (for fixed x > 4), which will allow us to apply Martins [ 11 ] solution of the following maximum problem. Choosing three energy values s 2 > s I i> s O > 4, we get, on account of eqs. (2.6) and (2.7), and because of the unitarity constraints
o < oo(x)
I,
(2.8)
the inequality
* We should mention that interesting mequahties on low-energy absorptive parts follow from the hmit t3 ~ O, after taking the appropriate number of derivatives [8, 9] (see also sect. 4).
P. Grassberger, 1t. Kuhnelt, Crossing sum rules
128
s2
(2t+ i) f /=2
sl
<~-5 f
o dx a2(x)aOO(x) I! - 5
4
+
dx a2(x) - C 1 .
(2.9)
s2
Our problem is now to find the maximal value allowed for an averaged total cross section, compatible with the constraints (2.8) and (2.9). The averaged total cross section is defined by $2
*rO*rO f f tot
---sl
*tono dxq(x) Otot (X)
= 32n G (21+ 1) f dx I=0 st
aOO(x),
(2 10)
with some normalized weight function q(x) to be determined later. The solution of this kind of problem is well known [ 11, 2, 4]. The maximum of -- ffOffO O tot IS assumed if the aOO(x) are given by 1,
a°°(x)=
for
0,
for
l<~L(x), t>L(x),
(2.11)
where L(x) is defined by x -4 x 4 OtL(x)(X) ~ ~ K "~ OtL(x) + 2(x) q ( x ) ' S2 f
L(x) dx~ ( 2 / + l)al(x)=-C I .
sl
(2.12)
(2.13)
1=2
Here, K ts some constant independent ofx. It is determined by eqs. (2.12) and (2.13). Numerically, the evaluation of the bound Is straight-forward. We used weight functions
q(x) -
const X6
with 0 ~< 8 ~< 2. The result was not very much dependent on the particular choice, and finally we settled on _ _
1
q(x) - x In (s2[sl)
(2.14)
P. Grassberger, tt. Kithnelt, Crossing sum rules
129
For the low-energy isoscalar d-wave, we used a modified Breit-Wlgner ansatz for the fo meson, which reproduced the correct scattering length [6] as well: F(s)2m 2
al2=O(s)=(mff _ s) 2 +
(2.15)
r(s)2m2'
with F(s)
2k
rf t(m - 4)(s + a)
a = 68.0,
~=
156 MeV.
(m 2 _ 4)-~ '
For the I = 2 d-wave, we used
a/=2(s) = k2 (k2(a + 4))4a/= 2 \
(2.16)
s+a
With these ansatz, we got C 1 ~ 1.06 × 10 - 6 (with s o = min (Sl, 200 m 2) , s I t> 120 m2). The resulting high-energy bounds, for s 2 = 10 Sl, are shown in fig. 1. For other ratios o f s 2 / s l , the bounds are shghtly higher. It is interesting that the bounds are only very weakly dependent on the low-energy input C 1. Thus, a change of C 1 by 30% always resulted in less than 10°6 change in the bounds. This is easily understood by the rapid increase of otI with increasing 1. It shows that improvements should not so much be searched by using more accurate low-energy input but by using some additional kind of information either on the lowor on the high-energy phases.
mb
j/ // /,11"
j~ t ~
50-
• S
Fig. 1. Upper bounds to energy-averaged norro total cross sections. The horizontal bars indicate the averagmg mtervals.
P Grassberger,tl. Kuhnelt, Crossingsum rules
130
Finally we should mention that asymptotically, our bound becomes equivalent to the Frolssart bound, o-- 71.O,/1.O tot ~ 7r( In s) 2. (The factor n in front cannot be improved without additional high-energy assumptions, in contrast to speculations in ref. [3]). Looking at fig. 1, one would suggest fitting a parabola o-~ono tot ~ const × (In s) 2, with const ~ 0.5 '< 7r. This shows very clearly that the curve shown m fig. I is not yet the parabola corresponding to the Froissart bound *
3. C h a r g e d
pions
For charged pions, one has two additional independent crossing sum rules [12]. Only one of them will be used here, tt is A2(tl' t2)--
×
f (X -
4
dx
tl)(X - t2)(X - 4 + t 1 + t2)
x44 Al(x, tl
AO(x, tt)-5A2(X, t l ) - 3
x + 2t 1 -
/
-[tl~t21
=0
(3.1)
Here, A I are the absorptive parts in the lsospm = I channels, and 0 ~< t 1, t 2 ~< 4. Since we want to constrain o ~r+'r° = ½(o 1 + o2), we need a sum rule getting high-energy contributions mainly from I :# 0. The one used in the following is (for A 1, see eq. (1.3)) 6A 1 - A2(4,0 ) + 4 ~ A2(t = 0,0) = ~ (21+ 1 ) ~ 1=2 1=0
l+l even
fdx~(x)a~(x)=O, 4
(3.2)
with 8
[p (x+4
'~
~l(x)-
x~(x_4){ L i ~ ] -
~(x)=3
3 x - 4 ~ I -x -~ 4x~(x- 4)2 3. . . .
/32(x)_ T--9 x - 1 6 --3
xT(x - 4)'i
21(l+ l ) x ] 1 + ~x ~ 4 ~ '
(x+4~
P / \ ~ - ]
[pl (kx~+_ 44 ~] - 1 1
-
1 -
x4_~(/2
(3.3)
+ 1-
2)
]
,
41(1+ 1)_ ( 9 x - 28). x{(x - 4)r
(3.4)
(3.5)
This combination is chosen because of the following properties of the ~3/(x): * After completion of this work, we found that similar results have recently be obtained by F. Ynduram, CERN-TH. 1554 (1972).
P. Grassberger, 1t. Kiihnelt, Crossing sum rules
(i) #t°(x) < O,
/3)(x)> O,
#2(x) > O,
131
(3.6)
for a l l l ~ > 2 , x>4; (i0 - ~ ( x ) , #}(x) and #~(x) are monotonically increasing with I (at fixed x > 4); 0ii) for l and x going simultaneously to infinity, m such a way that {x+4'~
et \ x : -~ / ~>O(x), we have
(x+4
~¢(x) ~ . ~ ( x ) ~ x -3 P l \ x _ 4 ,1'
(3.7)
while
#to(X) ~ X -4 el k x - - 4 - , 1 "
(3.8)
Since m addition, for fixed l, #~(x) ~ x -5 ,
(3.9)
we see that high-energy and large-/isoscalar contributions to the sum rule are suppressed. Indeed, one can use inequalities resulting from the crossing sum rule eq. (2.1) (ref. [9]; see also sect. 4) to show that already the contribution of the I = 0, l = 4 wave should be less than -~.of the contribution of the I = 0 d-wave. The higher I = 0 partial waves are completely negligible. We use eq. (3.2) by rewriting it as $2
~(21+ l=2 ~<
1) f dxl~](.;c)a](x)+~2(x)al2(x)] sl
dxll~to(x)laO(x)-C2 ,
( 2 l + 1) l=2
(3.10)
4
and use C2, the contribution of the isoscalar absorptive parts, as input. Because of the above-mentioned properties of #°(x), we can write (3.11) C2 < 7~X 5 f d x I~(x)la°(x) ~. 2.2 X 10 -5 4 (The situation concerning the higher isoscalar partial waves could partially be further improved by taking more refined combinations of the sum rules (2.1), (3.1) and a third sum rule not stated here [ 12]). Here we have again used the parametrization eq. (2.15). The further treatment parallels exactly the one for neutral p~ons, with the only exception that we now calculate bounds on o-'r+"° tot • The results are shown in fig. 2.
132
P. Grassberger, tl. Kia'hnelt, Crossing sum rules mb
'~tm
150-
/
/
/
/,d ...=e,./=(" _~/ic" =I=
50-
I:lg. 2. Upper bounds to n+n o total cross secnons. The bound is nearly the same as for neutral plons, except for rather low energies (s ~ 100 - 500 m2), where it is shghtly better.
4. Discussion and further improvements We hope to have clarified somewhat the role played by crossing symmetry, and especially the crossing sum rules, in deriving high-energy bounds. While the improvement over previous treatments [2, 4] was not very large in the low-energy domain, it was quite remarkable at higher energies. A direct comparison with the results of refs. [2, 4] is difficult since these authors always averaged over regions going down to threshold (i.e., they took s I --- 4). Furthermore, Roy [4] uses a weight function peaked strongly at low energies. For better comparison, we have redone their 7r°1r° calculations with our weight function (2.14) and s I = 2.4 GeV 2, s 2 = 24 GeV 2. We used a~ 0= 0.00073, and found a bound about 3 times as large as our hound. The results of Blankenbecler and Savit [3] are on the one hand less easy to compare since they used some extra assumptions on the high-energy cross sections. On the other hand, their method is similar to ours in taking into account also some lowenergy data from the physical region. Our approach seems, however, to be cleaner than theirs, which depends crucially on concellations in the low-energy input. These are automatically taken into account in the present work. For instance, our bounds are completely independent of the low-energy s- and p-waves. Improvements of our bounds could be made by either supposing some additional high-energy properties, or by changing the low-energy input. In the high-energy domain, one can assume e.g. the diffraction peak width, the ratio of total to elastic scattering or a constant cross section. What improvements can can be expected can be traced, e.g. from the review of Roy [4].
P. Grassberger, tL Kiihnelt, Crossing sum rules
133
On the low-energy side, one could assume the I -- 4-wave in addition to the dwave, and subtract its c o n t r i b u t i o n . However, this implies strong concellations between poorly k n o w n experimental quantities, and is subject to a similar criuque as ref. [3]. A cleaner, though less powerful, way is to use more i n f o r m a t i o n from the crossing sum rules. Let us sketch what one can do with neutral pions. Differentiating eq. (2.1) twice with respect to t l , and once with respect to t2, and taking t t = t 2 = t 3 ~ 0 at the end, one gets
dxTl(x)aOO(x) = O,
(2•+ 1) l=2
(4.1)
4
with
(x
x - (x - 4) { (x One sees that 7/(x) has similar properties as al(x ) 7l(x) > 0 ,
for
I/> 4 ,
72(x ) < 0 .
Indeed, the essential difference with respect to the a I is that n o w the Pl(z) are replaced by their Taylor expansions up to second order. As a consequence, eq. (4.1) does n o t lead to the Froissart b o u n d , b u t allows Oto t to increase like s~ -e, e > 0. For finite energies, however, using eq. (4.1) in the same way as eq. (2.4) above, we get results very similar to fig. 1 * A more complete m e t h o d would consist in treating eqs. (4.1) and (2.4) as simultaneous, i n d e p e n d e n t constraints. Finally, one can show that
~(x)-= st(x) .... vt(x),
•
has the following properties, a'2(x) < 0, ~/(x) > 0 for 1/> 4, and ~/(x) is m o n o t o n ically increasing with l. So, one can use also * For completeness, we should mention that these bounds are very similar to the bounds on resonance couphngs found In ref. [9]. For charged pions, we can shghtly improve a bound on the d- and f-waves [91 by using eq. (4.1) and a similar sum rule obtained from A2(tl, t2) (see eq. (3.1)) Subtracting these two from each other, we get
i x x,x 4,, , {.o,x>
6
Assuming here narrow resonance saturation, we see that
o >6(mf° 9 \mg /
} >o
134
P. Grassberger, tL Kithnelt, Crossing sum rules
(21 + 1) dx ~l(x)aOO(x) = 0 , (4.3) 1=2 4 as a starting point, instead ofeq. (2.4). Since on the one hand ~'2(m2o) "~ o~2(m2o), but ~/(x) ~- og(x ) for the values of/dominating the averaged cross sections at s 1, s2 -. o% one expects the bounds following from eq. (4.3) to be asymptotically better than the bound following from eq. (2.4). We have looked at this numerically. For the d-wave parametnzation gwen in sect. 2, we found that using eq. (4.3) instead of (2.4) indeed improves the bounds for s 1 ~ 400 GeV 2. For lower energies, this new bound Is however slightly worse: its lowest value, at s ~ 5 GeV 2, is 130 rob. We t h a n k Dr. D. Schwela for p o i n t i n g o u t t w o errors in the o n g m a l m a n u s c r i p t .
References [1] F.J. Ynduram, Phys Letters 31B (1970) 368, A.K. Common, Nuovo ('lmento A69 (1970) 115; A.K. Common and F.J. Ynduram, Nucl Phys. B26 (1971) 167. 12] A C. Steven, Cambrtdge preprmt IlEP 72--9 (1972). 131 R. Blankenbecler and R. Savlt, Phys. Rev. D5 (1972) 2757. [41 S.M. Roy, Physics Reports 5C (1972) 125. 15] M. Einhorn and R. Blankenbecler, Ann. of Phys. 67 (1971) 480; R. Eden, Lectures at Kalserlautern Summer School (1972) 16] D. Morgan and J. Pis6t, Springer tracts m modern physics, vol.55 (1970). 17] S. Krlnsky, Phys. Rev. 1)4 (1971) 1096. [8] G. Wanders, Nuovo Cimento 63A (1969) 108; GIFT-Seminar, 1971. [9] R. Rosktes, Phys Rev. 1)2 (1970) 247, 1649. [101 P Grassberger, Nucl. Phys. B42 (1972)461, P. Grassberger and I1. Kuhnelt, Nuovo Cimento Letters 4 (1972) 895 [11] A. Martin, Phys. Rev. 129 (1963) 1432 [12] P. Grassbcrger and H. Kuhnelt, to be pubhshed.