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Absolute bound on cross-sections at all energies and without unknown constants
Volume 31B, number 6
PHYSICS
ABSOLUTE
LETTERS
16 March 1970
BOUND ON CROSS-SECTIONS AT ALL AND WITHOUT UNKNOWN CONSTANTS
ENERGIES
F. J. YNDURAIN CERN,
Geneva,
Received
We prove that the total cross-sections
Switzerland
6 February
1970
for 711~ scattering
are bounded at all energies
by 3 zn(n!)
+ (1280)~IJ. < 7@$~log2~+ 8n(y) lJ2 1-L2 valid for all n > 1, and where u t is the t channel D wave scattering length. At high energies, hand side becomes essentially t?ie Froissart bound without unknown constants, viz., 1 j” ds’(+4p2)otot(sf) s-4/_l2 4p2
77[(ntl)/q2
(dp2)log2 (s/so),
the right-
s log2s /p-12.
The celebrated Froissart bound [I] for total cross-sections
atot(s) <
2 t [2n/(n+l)]n a 2 s, 12n) !
reads (for, say, 7171 scattering)
s>s
(1)
Here CL= pion mass, utot is the total cross-section and Gtot its average value; so and S are two unknown constants. The importance and great generality [2] of the bound, and the fact that experiment nearly saturates it, makes it desirable to remove its hindrances, namely, to obtain equivalent bounds without the unknown constant so and valid for all energies. In this note, we are able to solve the problem obtaining the absolute bounds, valid for any value of s,
n+l < lrn
2
( >slog22 P2
+ 8aq 1.12
\/=logs + (1280) B p k2 P2
c-4
where the bound is valid for any value of n 2 1, and ai is the t channel D wave scattering length*. For large values of s eq. (2) becomes +
4J
da’ (s’-4p2) atot (a’)
<,
?T(%)”
+log2+
,
(3)
which is equivalent to the Froissart bound. If, furthermore, one assumes that atot does not oscillate at infinity [4], one also gets by direct application of the techniques in the second paper of ref. 4, the local estimate (n -+ m) Gtot(s)
(
(cr+2) +
1og2+
(3’)
,
with @ = slLmmlog atot (s)/log (s/p 2) . *This is to be compared sip2
with a related
ds’ srm3 exp[2 log c
P2 368
result of Martin’s
4~ 2crtot (s’)/4n
log2 s-
P2
I]
(unitarity sum rule) [3] which bounds by
ai.
Volume
31B, number
PHYSICS
G
To prove our statement,
utot
(s)
=s%;
16 March 1970
LETTERS
eq. (2), we first recall
that* (4)
(21 + 1)77$4 ,
even A&,
-cc t) =s$f
(2Z+1)17&s)Pl
(5)
(cosQ,),
even and q = H 1q s-41-1 A, is positive when cos @, 2 1. where 7 z (s) = Imfl(s), cos Bs = 1 + 2 t/(s-4p2), Furthermore, by using the Froissart-Gribov representation, one can show [e.g. 51 that t channel D wave scattering length, ai = li2m f2(s)/q5, is given in terms of As by 4 -+o
t
1
ds s-~ As(s,
a2 = w
t = 4~‘)
.
(6)
From eqs. (6), (5), we obtain the chain of inequalities ds.s.-3Z~0 even
> $$%$2Z+l)
(21+1)~7~(s’)P~ (1+ %) 9-41.1
8Opq
>
4P2 S
s-2-l/2 >-P
J” ds’s’-3J’z(1+s~)~z(s’)
L+2
(
l+j&)
5
s-4p2
L+2
(2Z+l)_/2
ds’qz(s’)
(7)
.
If we split the sum in eq. (4) into two pieces, one from 0 to L = NC log (s/p2), and the rest (Nwill be fixed in a moment), multiply by ~-4~2 and integrate from 4p # to S, we shall get
-;$
4;
ds’ (s’-4p2)
atot (s’) = 32s
;
(2Z+l) ---&
s”
ds’ qz(s’)
+ s%2
4P2
even
z2(2Z+l) even
s” ds’qz(s’)
.
4P2
Using the unitarity bound, qI(s) c 1 for Z ,C L, and eq. (‘7) for Z > L, we obtain &
,$
ds’(s’-4p2)otot(s’)
< 32nN2-$og2
-+
+ 64aNr+log
$+
(Rest) 7
(84
where the (Rest) equals 32~ (Rest) c s-4p2
80 p qs2+v2 a “2
1 ~~+~(l + ~IJ~/[s-~P~])
’
@b)
Now, one has the bounds $, valid for all n 2 1, x > 1,
pz (4 >
(2n)f
[x+gqZ
,
2n(n!)2(2n/(n+l))n so that, with x = 1 + 81-r2/(s - 4p2),
1/PL+2 (x) is bounded by
* We are neglecting isospin. In fact, the complications 1 These bounds are due to A. Martin.
due to it are quite inessential.
369
Volume
31B, number 6
PHYSICS
-1 3c+ z&j-N@
(2n)I 2n(n!)2(2?z/(n+l)~
I
LETTERS
log(sh2)
16 March 1970
.
[
With a little bit of work, we then see that, in order to match the powers with the term (9), it is sufficient that
Since the bound (2) is anyway not very good for s near 4p2, this back into (9) and the result into (8b), we obtain the bound
(Rest)
<
3 2n (72!)2 [2n/(n+l)lfl (1280)~ ,u (2n) !
of s that appear in eq. (8b)
we can just take N = (n+1)/4n.
Plugging
t a2s ,
whence, substituting into (8a), the result, eq. (2), follows. It is interesting to ask whether this result could be generalized to other processes, say sN, KN or NN scattering where experimental results exist with which we could compare (2). This is certainly impossible at the moment if one insists in using only results provable from, say, axiomatic field theory, as the analyticity necessary for obtaining (6) has not yet been established. However, with some extra assumptions (essentially analyticity in domains corresponding to what field theory yields for ~7r), which are certainly fulfilled if Mandelstam analyticity holds, the analysis goes through exactly as before, and corresponding results obtain; the details will be presented in a separate paper. It is also to be remarked that in our derivation we have implicitly assumed that Martin’s pathology [6] does not occur, and ai is finite. Of course, the finiteness of ai has not yet been proved from “fundamental principles”, but it seems quite safe to assume it. Finally, let me comment a little on the quantitative value of the bound. If we take, e.g., (3’) (with cy = 0) for scattering in the range 1.5 - 10 GeV/c, the bound is of the order of - 2000 mb. Clearly, this is due to the fact that, in deriving the bound, we have taken the BITdynamics to be governed by 2s exchange, whereas experimentally it seems to be governed by the p. In fact, for pp scattering, and neglecting low-lying pion states, (3’) holds with thus bringing the bound to the order 21.( substituted by m,,. This gives a depressing factor of N l/100, of a few dozens of mb, i. e., the same magnitude as experimentally measured total cross-sections. It is also a priori clear that, say, (3’) cannot be improved too much as the constant factor is 2n/p2, only about above the geometrical limit of 2nr2, Y - l/(2 ,u)~. I would like to acknowledge a critical discussion with Professor A. Martin, to whom I also owe the M. Jacob and Dr. C. communication of the estimates for the Pl ‘s. Thanks are also due to Professor Schmid for pointing out the convenience of inserting the last discussion.
References 1. M.Froissart,
Phys. Rev. 123 (1961) 1053. 2. A. Martin, Nuovo Cimento 42 (1966) 930 and 44 (1966) 1219; H. Epstein, V. Glaser and A. Martin, Communs. Math. Phys. 13 (1969) 257. 3. A. Martin, private communication; Y. S. Jin and A. Martin, Phys. Rev. 135 (1964) B1375. 4. J. L. Gervais and F. J.Yndurain, Phys. Rev. 167 (1968) 1289 and 169 (1968) 1187. 5. A. Martin, Nuovo Cimento 47 (1967) 265. 1. A. Martin, in Problems of theoretical physics (Nauka Publications, Moscow, 1969).
*****
370
PHYSICS
ABSOLUTE
LETTERS
16 March 1970
BOUND ON CROSS-SECTIONS AT ALL AND WITHOUT UNKNOWN CONSTANTS
ENERGIES
F. J. YNDURAIN CERN,
Geneva,
Received
We prove that the total cross-sections
Switzerland
6 February
1970
for 711~ scattering
are bounded at all energies
by 3 zn(n!)
+ (1280)~IJ. < 7@$~log2~+ 8n(y) lJ2 1-L2 valid for all n > 1, and where u t is the t channel D wave scattering length. At high energies, hand side becomes essentially t?ie Froissart bound without unknown constants, viz., 1 j” ds’(+4p2)otot(sf) s-4/_l2 4p2
77[(ntl)/q2
(dp2)log2 (s/so),
the right-
s log2s /p-12.
The celebrated Froissart bound [I] for total cross-sections
atot(s) <
2 t [2n/(n+l)]n a 2 s, 12n) !
reads (for, say, 7171 scattering)
s>s
(1)
Here CL= pion mass, utot is the total cross-section and Gtot its average value; so and S are two unknown constants. The importance and great generality [2] of the bound, and the fact that experiment nearly saturates it, makes it desirable to remove its hindrances, namely, to obtain equivalent bounds without the unknown constant so and valid for all energies. In this note, we are able to solve the problem obtaining the absolute bounds, valid for any value of s,
n+l < lrn
2
( >slog22 P2
+ 8aq 1.12
\/=logs + (1280) B p k2 P2
c-4
where the bound is valid for any value of n 2 1, and ai is the t channel D wave scattering length*. For large values of s eq. (2) becomes +
4J
da’ (s’-4p2) atot (a’)
<,
?T(%)”
+log2+
,
(3)
which is equivalent to the Froissart bound. If, furthermore, one assumes that atot does not oscillate at infinity [4], one also gets by direct application of the techniques in the second paper of ref. 4, the local estimate (n -+ m) Gtot(s)
(
(cr+2) +
1og2+
(3’)
,
with @ = slLmmlog atot (s)/log (s/p 2) . *This is to be compared sip2
with a related
ds’ srm3 exp[2 log c
P2 368
result of Martin’s
4~ 2crtot (s’)/4n
log2 s-
P2
I]
(unitarity sum rule) [3] which bounds by
ai.
Volume
31B, number
PHYSICS
G
To prove our statement,
utot
(s)
=s%;
16 March 1970
LETTERS
eq. (2), we first recall
that* (4)
(21 + 1)77$4 ,
even A&,
-cc t) =s$f
(2Z+1)17&s)Pl
(5)
(cosQ,),
even and q = H 1q s-41-1 A, is positive when cos @, 2 1. where 7 z (s) = Imfl(s), cos Bs = 1 + 2 t/(s-4p2), Furthermore, by using the Froissart-Gribov representation, one can show [e.g. 51 that t channel D wave scattering length, ai = li2m f2(s)/q5, is given in terms of As by 4 -+o
t
1
ds s-~ As(s,
a2 = w
t = 4~‘)
.
(6)
From eqs. (6), (5), we obtain the chain of inequalities ds.s.-3Z~0 even
> $$%$2Z+l)
(21+1)~7~(s’)P~ (1+ %) 9-41.1
8Opq
>
4P2 S
s-2-l/2 >-P
J” ds’s’-3J’z(1+s~)~z(s’)
L+2
(
l+j&)
5
s-4p2
L+2
(2Z+l)_/2
ds’qz(s’)
(7)
.
If we split the sum in eq. (4) into two pieces, one from 0 to L = NC log (s/p2), and the rest (Nwill be fixed in a moment), multiply by ~-4~2 and integrate from 4p # to S, we shall get
-;$
4;
ds’ (s’-4p2)
atot (s’) = 32s
;
(2Z+l) ---&
s”
ds’ qz(s’)
+ s%2
4P2
even
z2(2Z+l) even
s” ds’qz(s’)
.
4P2
Using the unitarity bound, qI(s) c 1 for Z ,C L, and eq. (‘7) for Z > L, we obtain &
,$
ds’(s’-4p2)otot(s’)
< 32nN2-$og2
-+
+ 64aNr+log
$+
(Rest) 7
(84
where the (Rest) equals 32~ (Rest) c s-4p2
80 p qs2+v2 a “2
1 ~~+~(l + ~IJ~/[s-~P~])
’
@b)
Now, one has the bounds $, valid for all n 2 1, x > 1,
pz (4 >
(2n)f
[x+gqZ
,
2n(n!)2(2n/(n+l))n so that, with x = 1 + 81-r2/(s - 4p2),
1/PL+2 (x) is bounded by
* We are neglecting isospin. In fact, the complications 1 These bounds are due to A. Martin.
due to it are quite inessential.
369
Volume
31B, number 6
PHYSICS
-1 3c+ z&j-N@
(2n)I 2n(n!)2(2?z/(n+l)~
I
LETTERS
log(sh2)
16 March 1970
.
[
With a little bit of work, we then see that, in order to match the powers with the term (9), it is sufficient that
Since the bound (2) is anyway not very good for s near 4p2, this back into (9) and the result into (8b), we obtain the bound
(Rest)
<
3 2n (72!)2 [2n/(n+l)lfl (1280)~ ,u (2n) !
of s that appear in eq. (8b)
we can just take N = (n+1)/4n.
Plugging
t a2s ,
whence, substituting into (8a), the result, eq. (2), follows. It is interesting to ask whether this result could be generalized to other processes, say sN, KN or NN scattering where experimental results exist with which we could compare (2). This is certainly impossible at the moment if one insists in using only results provable from, say, axiomatic field theory, as the analyticity necessary for obtaining (6) has not yet been established. However, with some extra assumptions (essentially analyticity in domains corresponding to what field theory yields for ~7r), which are certainly fulfilled if Mandelstam analyticity holds, the analysis goes through exactly as before, and corresponding results obtain; the details will be presented in a separate paper. It is also to be remarked that in our derivation we have implicitly assumed that Martin’s pathology [6] does not occur, and ai is finite. Of course, the finiteness of ai has not yet been proved from “fundamental principles”, but it seems quite safe to assume it. Finally, let me comment a little on the quantitative value of the bound. If we take, e.g., (3’) (with cy = 0) for scattering in the range 1.5 - 10 GeV/c, the bound is of the order of - 2000 mb. Clearly, this is due to the fact that, in deriving the bound, we have taken the BITdynamics to be governed by 2s exchange, whereas experimentally it seems to be governed by the p. In fact, for pp scattering, and neglecting low-lying pion states, (3’) holds with thus bringing the bound to the order 21.( substituted by m,,. This gives a depressing factor of N l/100, of a few dozens of mb, i. e., the same magnitude as experimentally measured total cross-sections. It is also a priori clear that, say, (3’) cannot be improved too much as the constant factor is 2n/p2, only about above the geometrical limit of 2nr2, Y - l/(2 ,u)~. I would like to acknowledge a critical discussion with Professor A. Martin, to whom I also owe the M. Jacob and Dr. C. communication of the estimates for the Pl ‘s. Thanks are also due to Professor Schmid for pointing out the convenience of inserting the last discussion.
References 1. M.Froissart,
Phys. Rev. 123 (1961) 1053. 2. A. Martin, Nuovo Cimento 42 (1966) 930 and 44 (1966) 1219; H. Epstein, V. Glaser and A. Martin, Communs. Math. Phys. 13 (1969) 257. 3. A. Martin, private communication; Y. S. Jin and A. Martin, Phys. Rev. 135 (1964) B1375. 4. J. L. Gervais and F. J.Yndurain, Phys. Rev. 167 (1968) 1289 and 169 (1968) 1187. 5. A. Martin, Nuovo Cimento 47 (1967) 265. 1. A. Martin, in Problems of theoretical physics (Nauka Publications, Moscow, 1969).
*****
370