SCATTERING and MESONS
by
FETHI MUBIN RAMAZANOGLU
Submitted to the Department of Physics in Partial Fulfillment of the Requirements for the
Degree of
BACHELOR of SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 2006
© 2006 FETHi MOBIN RAMAZANOOLU
All Rights Reserved
The author hereby grants to MIT permission to reproduce and to distribute publiclypaper
and electronic copies of this thesis document in whole or in part.
/
m
Signature of Author
>$/epartment of Physics
A/
Certified
Accepted by
Y.
-
A
Y-C.Y
/
/ ' 7,af/rofessor
Robert L. Jaffe
ThesiSup
isor, Department of Physics
V
Professor David E. Pritchard
SeniorThesis Coordinator, Department of Physics
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OF TECHNOLOGY
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0 7 2006
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Scattering and Mesons
F.M. Ramazanoglu
MIT Department of Physics and De artment of Mathematics
May 12, 206
Abstract
We present the P-matrix, an alternative method to parameterize the S-matrix,
which is particularly useful for low energy meson-meson scattering. We discuss
its basic properties and use it to analyze the isospin 0 and 2 s-wave r7r scattering. We construct the S-matrix from our analysis and discuss the physical
relevance of its poles. Further details of P for more general cases are provided
in the appendices.
1
Introduction
With the development of QCD four decades ago, many of the strong interaction phenomena could be successfully explained, and the strength of the theory was enhanced
with the subsequent discovery of many proposed bound states of quarks. Nevertheless, two body scattering at low energies is still a problematic issue. All our scattering
experiments are performed on baryons and mesons, systems of bound quarks, instead
of the individual quarks and this makes the calculations and analysis of data quite
hard.
Traditionally, we parameterize the S-matrix in a way to ensure its unitarity and
there are infinitely many ways of accomplishing this, but by far the mostly employed
technique is the K-matrix. As long as the we are concerned with narrow resonances,
our choice of parameterization does not affect our analysis significantly, but low energy
scattering is full of broad structures, r7r isosinglet s-wave below 1 GeV and the
small negative phase shift in isospin 2 s-wave being two basic examples. Clearly, our
choice of parameterization should go beyond simply following the tradition of K and
2
F.M Ramazanoglu
reflect the physics behind these processes if we want to effectively analyze these broad
structures in the scattering of quark systems.
We present an alternative parameterization of S: the P-matrix parameterization,
which can be more suitable for the non-resonant structures we mentioned. The reason
behind our choice of P is that it reflects a fundamental property of quark systems,
the fact that they are strongly confined within a volume. An investigation of the
physical meaning of the other well known parameterizations, such as K shows that
the specific reasons they were initially constructed are not relevant to low energy
meson scattering. By using P, we believe we can use our knowledge about the physics
behind the scattering process more effectively while analyzing the data.
We start with a basic discussion of scattering and than introduce P together with
other parameterization methods. Following this, we show how to parameterize P
itself and illustrate how to use it on isospin 0 and 2 s-wave rir scattering. Lastly, we
construct S from the P-matrix that we obtained from the data analysis, and comment
on its success to explain physical resonances.
2
The Boundary Matrix Method
We begin with a discussion of the basic mathematical tool of scattering experiments:
the S-matrix. Most of the results we present about it can be found in any elementary
quantum mechanics text (for example, see Ref. [1]). We demonstrate different ways of
parameterizing S and introduce our main point of discussion in this treatment, the
P-matrix parameterization.
We explain the physical significance of different parame-
terization methods and give the reasoning behind our choice of P.
2.1
S-Matrix and Its Poles
Consider a state
(E), in a general scattering experiment where a particle interacts
with a potential. If we denote the free state in the distant past that evolved into
this state by Tin(E); and the free state that this state will evolve into in the distant
future by T,,t(E), then scattering matrix is defined as
out(E) = Si,,(E)
(1)
We call S a matrix since we may have many different possible incoming states and
many different outcomes (called channels) in a scattering experiment of particles,
which make TAsinto vectors that are related through S.
3
P-Alatrix Theory
For radially symmetric finite range interactions in 3 dimensions, the equation for
the radial wavefunction is
d2
dr 2
where u = r(r)
k2 +
-2
- V + 1(1
)U=,
r2
and I is the angular momentum number. In our treatment, we
will solely analyze the radial component of the s-wave, i.e. 1 = 0 case. Under these
restrictions, the problem is reduced to that of a 1D case with the same potential
but where the 1D wavefunction is replaced with u, the coordinate axis is the radial
distance r and there is the boundary condition u(0) = 0. Treatment of the higher
angular momentum cases can be found in the appendices.
Outside the range of the potential, the solutions for the s-wave are proportional
to eikr.
The - solution is an incoming wave and the + is the outgoing one. If we
send a plane wave(e- ikr) to a finite range potential, final state outside the range can
be written as
u oc Sekr - e-ikr
(2)
neglecting overall factors. Relative phase is chosen such that as the potential vanishes
and S -- 1, the state satisfies the boundary condition at r = 0.
A short calculation of probability flux shows that
Is1 = 1.
which enables us to write
S = e2i5
where 6(k) is called the phase shift. The factor of 2 is to make further calculations
shorter as we will see.
In experiments, we can only measure real values of k; but in analyzing the data,
analytically continuing S into the complex plane and analyzing the structure of its
poles give valuable insight to the physics of the system. Poles of S on the imaginary
axis are easily interpreted and have physical importance, so let us start with analyzing
k = i for real . Clearly, ei(i)r = e-"r term dominates as S - oo. For > 0 this
is a decaying exponential which indicates the behavior of a bound state outside the
range of classically allowed region. For ri < 0, we encounter a less familiar state, the
solution outside the range of the potential is exponentially growing. Although it is a
solution to the wave equation, it does not satisfy the additional requirement of square
integrability.
These solutions are called virtual states. Note that for both cases, in
4
F.M Ramazanoglu
a classical system we have negative energy which is indicative of a bound state since
_ 2m
E
'
Although the imaginary axis poles are physically meaningful, they represent only
a very small region of the whole complex plane. Often, S has poles off the axes and
in this case, we can have physical insight by considering the corresponding energy
which will have a complex part
E = Eo- ir.
If we insert this into the time evolution operator e-iEt
= o(e-i(Eo-ir)t)o 1l = e-rt
we see that this state is decaying for F > . We call such states resonances. We
should bear in mind that F < 0 gives an exponential blowup in the probability which
is not physical. Since energy is related to k 2 , the sign of r is the opposite of the sign
of the imaginary part of k. This implies that resonances occur below the real k-axis.
To sum up, there are three significant types of poles: bound states, virtual states
and resonances. There are additional criteria to assess the significance of a pole in
the complex plane and we will elaborate on these in the following sections.
2.2
Parameterizing the S-matrix and the Logarithmic Deriv-
ative
As for any data, we need to presume a certain functional form for S in order to
analyze the scattering experiments. For example, one common strategy is using the
K-matrix defined as:
1+ ikK'
and K itself is parameterized as a sum of poles
K =_
2
k21_k+
(4)
This way of parametrization emphasizes our analysis of the physics in terms of the
poles. The data analysis is basically deducing the parameters Al, kl, etc.
It may be noted that this strategy is not exactly what we explained in the beginning; rather than asserting a functional form for S, we first linked it to another matrix,
P-Mlatrix Theory
5
ox
L
1.5
_'
a
.
I ' ' I . . . .:. . .
.
... .. ...
bound
state·····
: ... ... ... .
........... ·
........................
.. . a '
a
r
0.5
_.
. . .
. . . . . :. . .
. . . .. . : .. . ... .
a
U
E
-¢ IS
. . .. . . ... .. ... . .
.
.
.
.
........... :..
....
.. .
resonance .
: virtual
..................................................................
-1
-
................
1.5 ................................................................
-l
-2
i
-1.5
i
i
-1
-0.5
0
0.5
.................................
1
1.5
2
Re{k}
Figure 1: Possible locations of the poles of the scattering matrix and their physical
meanings.
K, and parameterized K. This is a technical necessity to satisfy the requirements for
S, mainly the unitarity condition. It is difficult to directly impose unitarity to some
random functional form of S. In the K-matrix approach, S is automatically unitary
if K is hermitian, and hermiticity is an easy constraint to satisfy for the functional
form we proposed. We will always use an intermediate matrix for parameterizing S
in our discussion.
Although most widely used, the K-matrix parameterization is by no means the
only possible path, and our main discussion will be another parameterization, the
so called P-matrix. Originally proposed by Jaffe and Low in Ref.[2], P is defined
through the logarithmic derivative of the wavefunction at a fixed point b:
P
(k, b)'b)
(n((k, b)))'= ,O(k,
First, remember that
C e2iteikr - e-ikr
(5)
F.M Ramazanoglu
6
from eq.2 and the fact that S is unitary.
Picking a distance b outside the range of the potential,
ike2 i eikb- +
+-e-ikbp
ike-kb
e2i eikb
=
e2ieikb(P- ik) = (P + ik)e-i kb
2i6
-ikb
=
1 + i k,
-ikb
(6)
or
P = k cot(kb + 6(k)).
(7)
The first straightforward observation on P is that it has a pole when the wavefunction vanishes at r = b. Before investigating the physical significance of this, let
us introduce a closely related parametrization called the R-matrix, formulated as
R
e
-
(ln(4'))'Ib
=
4"' |(8)=b
S = e- i k b 1 - ikR
1 + ikR
-ikb
(9)
or
R = -tan(kb + 6(k)).
(10)
R is the inverse of P and has the property that it has poles when the first derivative
of the wavefunction vanishes at r = b.
R was first introduced in Ref. [3] to explain very low energy scattering off the
nuclei, which explains the relation between its poles and vanishing derivative. Assume
nuclear interaction occurs in a finite region of the space with radius less than b, so
for r > b we have a free wave solution with a large wavelength (compared to b).
In a scattering experiment, as long as the first derivative of the wavefunction is
nonzero, the amplitude of the scattering wave outside the potential will be very large
compared to the amplitude inside the potential (since the wavelength is big), so what
we approximately observe is a wavefunction vanishing at r = b, which corresponds to
the hard sphere scattering for which 60 = -kb. The significant regions of scattering,
where +'(b) is small, show up as jumps, when the derivative of the wavefunction at
r = b switches sign. If the size of the interaction range is exceedingly small, than
we can simply say that wavefunction vanishes at the origin, i.e. 6 = nr. Since the
significance of scattering is deduced from the scattering cross section, which behaves
as a sin2 6, the scattering is insignificant unless 4'(b) is near zero.
P-Mfatrix Theory
7
When the first derivative of the wavefunction vanishes at b, than we observe a
phase shift of 6 = (n + 1)7r/2. This corresponds to strong scattering since sin2 6
1. Overall, these observation shows that poles of R reflect the values of k where
significant scattering occurs.
From eq.3 and eq.9, we see that K is the limit of R as b - 0. To put it in more
physical terms, K is a suitable parameterization when the size of the interaction
region is very small compared to the wavelength of the scattering particles.
Poles of P correspond to the vanishing of the wavefunction at r = b, which is
an attribute of confinement; and this is the main reason behind our choice of this
parameterization for meson scattering. Meson scattering is in fact a scattering of
systems of quarks and confinement is an important feature of quark systems. Our
current understanding of bound quark systems is not complete, but their confinement
is a well known notion. P reflects this important notion, even if not perfectly, and is
more desirable for our analysis compared to some random parameterization. Following what we said previously, R and K are clearly not desirable approaches to meson
scattering. For the data we are going to examine, wavelength is comparable to the
size of the scattering region, which undermines the basic physical motivations of R
and K.
Lastly, we should emphasize that these are only a few of the infinitely many ways
to parameterize S. A very clear example is the M-matrix parameterization which
again uses the logarithmic derivative of the wavefunction and is itself parameterized
by a real number in the interval (-7r/2, 7r/2]
M(0) = tan(kb + 6(k) + 4),
(11)
where / = 0 corresponds to R/k and
= r/2 corresponds to -kP. It is not
straightforward to see what kind of physics will be related to M for the intermediate
values of . Using trigonometric identities and naming L = ,X
M()=
L + tan q
1- Ltan
(12)
This shows that, for intermediate values of 0, M(0) has a pole when the logarithmic derivative of the wavefunction is equal to some nonzero, finite value. For
the cases of P and R, L was zero or infinite, which had simple physical meanings,
but the situation is more complex for random . Nevertheless these intermediate
parameterizations
remain as options, at least mathematically.
F.M Ramazanoglu
8
3
Scattering off the Square-Well Potential
In this section, we closely examine the scattering off a finite square potential well,
which will have a basic role in our subsequent model. Then, we introduce the basic
mathematical structure of P and show how we can modify P from its no-potential
form to obtain an expression for general potentials. This modifications turn out to
be subtler than the case of K and we will discuss the physical meaning of them.
3.1
Scattering and S-matrix Poles for Square Well
In general, we can calculate the S numerically if we know the potential, but analytical
expressions are available for certain cases. The square well is a particularly important
case since the physical motivation for the P-matrix parameterization is closely related
to it, so we are going to discuss its physics in detail. We will follow Ref.[4].
Consider the potential
V(x)=
-VO r < b
When we solve the Schrodinger Equation, the solution outside the well, up to normalization, is
b(r)= ei6 sin(kr + 3) r < b,
and inside the well is
+b(r)= Asin(qr) r > b,
where
q2 = k2 + ko2
k2 =
2mV
h2
Then it is straightforward to calculate P:
P(k,b) = qcot(qb).
(13)
We have all the information we need about P in eq. 13 and we will go on working on
it in the next section. For now, let us concentrate on S.
The phase shift is given by eq.6
e2i = e-2ikb
1 - i k ct
b
cot qb(14)
and it has poles when
y cot y = -iz.
P-Matrix Theory
9
qn
Figure 2: Solving the energies of the finite well. Intersections of the semicircles
and -ycoty correspond to the solutions. As the semicircle grows in radius, the
intersection points move.
where we use the more convenient variables z - kb, y - qb and zo - kob.
When k is real, so is q, and there is no way the denominator can vanish since the
factor that multiplies i is real. So we have no poles for real k.
To look for poles on the imaginary axis, we set z = i, and restrict our attention
to the cases where r7and y are both real. Then qris real and r,12 < z02, so the condition
for a pole becomes
ycoty = -r7
y2+72
=
z2.
This corresponds exactly to the condition for antisymmetric bound states of the
finite square well (for a discussion of the bound states of the finite square well, see,
for example, Ref.[5]). Thus, we conclude that we have poles at the energies of bound
states.
F.M Ramazanoglu
10
z=4.5
-5
0
zo=5
5
10
-10
-5
Zo=5.5
0
5
10
=6
'A
9
Re{kb)
Figure 3: Position of the poles of S as the circle cuts the second arm of -y cot y
(see FIG.2). Top left: Just before intersection. There is only one bound state and
two off-axes states are near k = -i. Top right: Two close off-axes poles disappears
and two virtual poles appear and move in opposite directions, one becoming a bound
state. Bottom left, bottom right: Virtual states and the bound states move out as
the off-axes poles move in (the largest two circles in FIG.2).
This analysis is not complete since we do not know whether all poles are at the
bound state energies, and from FIG.2, it is evident that this is not the case. While
finding the bound energies, we had not considered t7 < 0 half, but these also give
poles in our graphical analysis. The situation becomes clear if we examine the wave
function outside the well for imaginary k, k = in:
ei sin(ilr+ ) oce- mr+ e-2i e,r
which reduces to e- Nr for a pole in S. Now we can see that
> 0 gives a dying
exponential, as in the case of bound states; and n < 0 gives exponential blow up,
which are the virtual states we previously discussed.
P-Matrix Theory
11
To have a better understanding of the poles, let us consider what happens as we
deepen our potential, which corresponds to circles with growing radii in FIG.2. For
small radii, we have no intersection at all and the first intersection appears when z0
reaches 1 and = -1. As the radius gets larger, the intersection point moves and
after z0 = r/2, it is on the > 0 half plane in FIG.2. Remembering that = -ikb,
we see that we start with no states, than have a virtual state that moves as the
depth of the potential increases, and finally a bound state. After this point, nothing
interesting happens for a while , until the circle touches the second arm of y cot y
around z0 = 4.6. As we continue to enlarge the circle, it intersects the second arm of
y cot y at two points. One of the intersections moves towards positive rl and eventually
becomes a bound state as in the previous case. The other, on the other hand, moves
towards more negative 7 and remains a virtual state. As we increase the the depth of
the well more, situation continues similarly. For deep enough potentials, we always
have both virtual and bound states.
Poles of S in the complex plane can be seen on FIG.3, the motion of bound and
virtual states are clearly visible. Interestingly, there are symmetric, off-axes poles that
we have not mentioned, which seem to move closer to -i as the well deepens and
disappear when a virtual state appears. We will not attempt an analytical discussion
of these off axes poles; basically, there are infinitely many of them. We should keep in
mind that not all off-axes poles are resonances, and in order to have physical meaning,
these poles should be sufficiently close to the real axis.
3.2
Compensation Poles and Parameterization
An important limit for a parameterization
which, for P, corresponds to:
matrix is the vanishing potential case,
lim P(k, b) = k cot(kb).
Vo--0
This limit of P, which we willrefer to as the "compensation P", has the interesting
property that it has poles, in fact infinitely many of them, at kb = nr for n =
1, 2, ... Complexity of this structure maybe better appreciated when we note that the
compensation K is simply 0.
The rich structure of the compensation P has a significant non-trivial implication
for our subsequent data analysis: If we try to implement the intuitive idea that
scattering off a potential can be understood by modifyingthe compensation P through
F.M Ramazanoglu
12
some parameters, our starting point should be modifying the function k cot(kb), rather
than simply adding poles to P = 0. The physical origin of these infinitely many poles
lies in the fact that even for the non-interacting case, there are infinitely many k
values for which the wavefunction vanishes at r = b. For the compensation P, Po,
PO
k
-
= cotz =
1
0
Z
n=1
(1 + 22ZE2
Pz
1
22
222
2z _n2 7r
. /
Existence of our scattering potential has the effect of moving the poles of P from
their original positions of nir and changing the residues. The easiest way to implement
this is to completely remove a pole from the compensation P and add a new one
2r
2
11
87r2
72
bP=zcotz- [+ 2-r 2 z2 -4r2 ] +C+z 2 2-+ -2
+
(15)
Note that we also let the constant term change.
For some cases, our analysis may need poles that do not exist in the compensation
case at all, here we simply add a pole to the compensation P
kPl(16)
k2-2'
=
Po
+
but we will not need such a modification for the cases we analyze in this discussion.
In general, we both move poles and add new poles.
The main two modifications, moving and adding poles, have distinct physical
meanings. Pure modification of the poles represent the infinite tower of Q 2Q 2, and
adding new poles is interpreted as QQ(see Ref.[6] for details).
Eq.15 incorporates everything we have mentioned about P. Once we have the
data, (k), from our scattering experiment, we can run our preferred computer pro-
gram and extract the values of C, y, zl, Y2and z2 , using P = kcot(kb + 6(k)).
We are almost at the point where we
that our sole duty is finding the best curve
k. This approach is ignoring the fact that
shift. For example, the best known one is
can analyze our data, but Eq.15 assumes
that fits 6(k) for the experimental range of
there are general constraints on the phase
the low energy behavior of 6 which is
1
lim k cot = -
(17)
a
where a is called the scattering length and sometimes can be calculated experimentally
k-O
or theoretically. Assuming we know a, it is easy to formulate C in terms of a and the
other parameters we use:
C=+a+
- 2) + 22-2)
(18)
P-Matrix Theory
13
where a = a/b. This way, we can write the constant term in terms of the other fitting
parameters and have one fewer degree of freedom.
Analyzing 7rr Scattering for I = 0 and I = 2
4
Through P-matrix
In this section, we perform the main calculations with P, whose basic properties we
have presented up to now and in the appendices. We apply our results to isospin 0 and
2 pion-pion scattering data. Using only three parameters (moving two P poles and
changing the constant term), we can extract the parameters of P with high accuracy
(in terms of reduced X2 ) and low uncertainty.
After extracting P, we construct S for the complex plane using eq.6 and try to see
if the poles correspond to physical resonances. One major aspect of S we investigate
is robustness under small variations of the parameters of P, and S seems to be robust.
However, S has an infinite array of poles that are not related to the resonances we
know. We comment on the origin of these poles and discuss how to interpret them.
4.1
Deducing The P-matrix parameters
In our subsequent analysis, our data for s-wave r7r isospin 0 scattering is from Ref. [7]
and isospin 2 scattering is from Ref.[8].
For our analysis, we slightly modify our fitting method in eq.15 so that the quality
of the fits and the locations of the poles are more visible. In the subsequent treatment,
we fit the parameterized function
F(y, z, 72, z2) = z cotz-1-
8712
27r
2
2
z - 7r
z
2
-
7 + Z2
4 2
71
-
Z
+ Z22 -
2
7
z2
1Y
+ 72
+
z
z2
(19)
to the data
D = zcot(z + 6) -
+ 2 +2.
where z = kb as before and b = 1.4 fm.
First, we apply our method to I = 0 7rirscattering, without taking into account
the low energy behavior of the phase shift. A preliminary analysis where we moved a
single pole gave C = -1.23+, 17, 7 = 5.39 ± .55 and zl = 2.063 i .012, which gives
14
F.M Ramazanoglu
60
.\.......................................
.........
......S .....
data
\
:
fitted curvel
40
20 ... . . : .
.
. ...i ............. .
0
. ................................. ......................... .................
-20
....... ............ ..... ...
-40
... ... .. ... ... .. .. ... ... .
. . . . . . . . . . . ... . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . .. . . .. . . .. . . .. . . . . . . . . . . . ...... ..................
..........
~~~~~~~~~..
-60
2
2.1
2.2
I.
............................... ........................ ..........................
2.3
.. . .
.
..
.
2.4
.. .. .. ..
.
2.5
.
.. ......
....
2.6
........ ......
2.7
.
.
2.8
. .......
.. .....
2.9
3
z=kb
1.6
-
. .
..
.
.
.
.
.
.
.. I....
0I-
0.25
.
0.3
0.35
0.4
0.45
CoM momentum
k (GeV)
Figure 4: Upper Plot: Fit of F(yl, z1, 2,Z2) (eq.19) to the data D for I = 0 and for
moving two poles (three parameters) with the second residue fixed. Lower Plot: 6 vs
k for the parameters in the upper plot.
a X -3, well outside the known range of a = 0.216 ± 0.022m; 1 or
= 0.225 (from
Ref. [9]).
Incorporating a into our model is not straightforward, since the known values have
a relatively high uncertainty. To overcome this, we used a range of different values
(a = 0.1 to 0.4) and checked the Reduced x 2 - X values of the fits for the whole
range. For all fits, the quality of our fit (X2) does not vary much with a, so we can
safely pick the value of a = 0.225 and continue our analysis.
Once we know how to handle the k - 0 behavior, we should decide how many
poles in Po we should modify. Our initial attempt of moving a single pole (thus two
parameters, location and residue of the pole) gives a high X2, but since it is not very
far from the ideal value of X2 = 1, we can see that we do not need to move many
poles (see Table.1). Results for moving the first two poles (4 parameters) are also in
Table.1. This time, the X2 is much below one. Thus, with only two trials, we can
P-1atrix Theory
15
r
.........
5 .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . .
L.
..... ..
o
-5 ..
q~
_1[.
-
.....
.. .... .. ......
. ..... . . ...
...
.
1
1.5
2
2.5
3
3.5
.-
I
4
:.
r
-. .
data
I ..- I
i
fitted
I
5
4.5
-
z=kb
-0.2
~~~~~~
,..................
~
~ ~~ ~
'0
-0.4
E-
'o
-0.6
-0.8
0
.. . . . . . . .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
. . ..
L.
. . . . . . . ..
.. . . ...
0.5
......
..
..............
. . . .. . . . . . . . . . .
1.5
CoM momentumk (GeV)
Figure 5: Upper Plot:Fit of F(yl, z, y2,z 2) (eq.19) to the data D for I = 2 and for
moving two poles (three parameters) with the second residue fixed. Lower Plot:6 vs
k for the parameters in the upper plot.
conclude that a 3-parameter fit would be ideal.
There are two simple approaches to a 3-parameter fit: We fix the second pole
location, having the second pole residue as a parameter; or we fix the second pole
residue and have the pole location as a parameter. The values from both analysis are
in Table.1 and a plot of the latter is in FIG.4. Again, our X2 values are below 1, but
counting the small positive contribution that would arise from the uncertainty in a,
the fit is quite successful.
Isospin 2 r1r scattering is very similar to the I = 0 case. Fitting procedure for
the P-matrix is exactly the same, except we change the mass of the meson to that of
m,+ =139.6 MeV and the scattering length to -0.0444m; 1 ( from Ref.[10]).
A 3-parameter fit again turns out to be the most appropriate in terms of XR (see
FIG.5). Although statistically satisfying, our curve for I = 2 has an undesired behav-
ior outside the range of our data (k > 0.8GeV in FIG.5). We would expect a more
F.M Ramazanoglu
16
gradual decay to 0 rather than having a "bump". This is probably a mathematical
artifact of our parameterization scheme, but we will use the results for the following
sections.
Table 1: Summary of results for I=0 and I=2 data analysis for different numbers of
parameters.
I=
#parameters
Pole-1 location
Pole-1 residue
Pole-2 location
Pole-2 residue
X2R
4.2
0
0
2
4
2.076 +t .022 2.054 + .008
2.963 ± .215 4.156 + .240
3.488 t .158
6.874 t 3.256
2.57
0
0
2
3 (fixed location)
2.058 t .010
4.879 + .284
239.2 + 22.4
3(fixed residue)
2.057 + .010
4.785 + .240
5.002 t .082
-
3(fixed residue)
3.51 ± .03
26.3 ± 1.0
7.15 ± .30
.46
.40
.83
.23
Constructing and Testing the S-matrix
Ultimate aim of our analysis is to understand various bound states and resonances
from the data. P has an intermediary role in this process, such that we can construct
the scattering matrix S once we know it; through
= -e2ikb1 - iP/k
(20)
1+ iP/k'
Before analyzing S on the complex plane, let us review one point about the poles
of S for non-real k. All our experiments, the source of all our data, are on the real
k-axis. Thus it is not justified to attribute a physical meaning to poles that are
substantially away from the real line and do not have a considerable effect on it. So,
our focus will be on the poles near the real line.
Let us start with constructing S for the fixed residue three parameter fit to I = 0
case, where y1= 4.785±.240, zl = 2.057+0.010, z 2 = 5.002+0.082 and uncertainties
correspond to one standard deviation confidence interval. Results are in FIG.6.
Following our guideline about the off-axes poles, first two nearest pairs of poles
symmetrical around the imaginary k-axis, are distinctly isolated from the rest. We
will elaborate on how to interpret these in the conclusions section.
For isospin 2, we follow the same procedure, plot of ISI is in FIG.7. In the I = 2
P-lMatrix Theory
17
-
-1.
i
-2.
-3.
2
4
6
8
10
12
14
16
18
Reb})
Figure 6: General view of the significant poles in ISI for the T7rr,I = 0 three parameter
fit. Darker areas indicate higher values, thus poles. Note that the leftmost two poles
are very near to each other, but are distinct.
case, one of the poles is again significantly further from the others and nearer to the
real k-axis, but its significance is less apparent compared to the I = 0 case. Again,
there seems to be an infinite array of poles, and we do not have a physical reasoning
to rule out or include any o them as physically relevant.
It is not hard to trace the origin these unexpected poles, this infinite series of
poles resembles the infinite-pole structure of the compensation P. Compensation P
has an infinite array of poles, but when introduced into the equation of S, poles in the
numerator and the denominator cancel each other to give unity in the whole complex
plane. Although cancellation is exact for the compensation case, it turns out that
even very small deviations from the compensation P results in an infinite array of
poles of S. This effect is clear in FIG.8, where we slowly move the location of the first
pole in PO, without making any other changes, in the interval 3ir/2 > z > 7r/2. Even
small changes give rise to an infinite number of poles of S in the complex k plane.
18
F.M Ramazanoglu
0
-0.5
.....................
unnunnn:
1 :1 1 :1 1 1 :1 1 :1 1 1 ........ :............
.....................
.....................
.....................
....................
............................
.....................
....
.....................
unnnunnu
.....................
......
.....................
.............................
...................
..
. . . . .. . ..
..
. ... .....
..
..
..
..
..
..
.
..
-1
a
r -1.5
-2
.
.
.
.
.
.
.
.
...
...
.. .
.. .
...
.. .
.. .
...
.....
.....
.....
.....
.....
.....
.....
.....
. . .. .. . . .. .. .. . . . .. ..
.. .... .. .. .........
.....................
.....................
.. .. .... .. ..
.. .... .. .. .........
.. .. .... .. .. .........
.....................
.. .. .... .. .. .........
.. .. .... .. .. .........
.....................
.. .. .... .. . .........
.. .. .... .. .. .........
...... ..
...... ..
.. . . . . . .
.... ..
.. . . . . . .
...... ..
...... ..
..... ..
.............
......
............
.....
...............
....................
.......................
....................
.............................
................
.... .................
..................................
...................
.....................
......
.............
............
. . . ... ... . .
. . .
... ..... ....
... ..
... ..... ....
... ..
... .....
... . ... . . .... . . ..
. . . . . . . . . .. . . . .
.. . . . .
.. .
. . . .. . . .
.. ..
.. . . . . .
.. . . . .
.. ..
.. ..
.. . . .. .
.. . . .. .
... .. ..
. ......
..... ......
... ... .......
...
...
.. .
.. .
.. .
...
...
...
...... ..
...... ..
...... ..
. .. . . .. . ..
. . . .. . . .
. . . .. . . .
... ..
.....
. ..... . ..
.. . .
.. . .
.. . .
.. . .
.. . .
.. . .
.. . .
.. . .
. ......
I ...
.. .. .
.. .. .
.. .. .
.. ..
.. . .
. . . .. ..
. . . .. ..
. . . .. ..
.. ..
......
. .. . . .
. .. . . .
. .. . . .
. .. . . .
......
. . . .. . .
. . . .. . .
. . . .. . .
-2.5
0
2
4
6
8
10
Re{kb)
12
14
16
18
Figure 7: General view of the significant poles in SI for the ir7r,I = 0 three parameter
fit. Darker areas indicate higher values, thus poles.
The conclusion is that the infinite array of poles is an artefact of our mathematical
fitting model.
We should add as a side note that inappropriate handling of the truncation errors
may also lead to poles in the exact compensation case, we were careful to avoid such
computational errors.
There is one more aspect of S to test: Robustness. Our values for the parameters
of P unavoidably have uncertainties attached to them, so plotting ISI only for the
central value of the parameters may be misleading. After all, we do not want the
general structure of S, that is the locations and residues of its poles, to change
significantly when we move within the interval of uncertainty of a parameter. If this
condition is not satisfied, our interpretation of the poles as resonances are unjustified.
Let us again concentrate our attention to the three-parameter, second residue
fixed P-matrix fit. If we change our fitting parameters slightly, than the locations of
the poles in S also change. If poles of S do not move drastically than we can say
P-lMatrix Theory
19
-3
-4
-5
-6
-7
0
2
4
6
10
8
12
14
16
18
ReO(b
Figure 8: Poles of ISI as the location of the first pole in P is moved in the interval
37r/2 > z > 7r, all plotted on the same graph. Movement of the poles is in the
directions of the arrows as zl becomes smaller. An infinite array of poles appear even
for the slightest change in P.
that our method to construct ISI is robust. By a slight change in the parameters, we
mean changes at the order of a standard deviation. In FIG.9, we can see how the two
leftmost poles in FIG.6 move as we move on the first standard deviation ellipsoid in
the space of the fitting parameters, which is
('
-
1)2
+ (zI
- Z)
2
(
+Az
- 2)2
=
where the barred quantities are the central values from our P-matrix fit and the A
are the standard deviations of the parameters. We can see that, poles move only
slightly as we change the parameters. Thus, S we constructed is robust for the case
of I = 0 s-wave scattering.
Although it is hard to investigate robustness analytically for a general pole off
the real axis, this task is relatively easy for the poles on the imaginary axis, which
F.M Ramazanoglu
20
l
iF
1100
Z
'r
-Zb
.t-
Z.
-.
-Z
-1.
-1 .b
-1.4
Re{kb}
Figure 9: Absolute values of S as we move on the first standard deviation ellipsoid,
drawn all superimposed on each other. The bigger and tilted lower ellipse roughly
contains the locations of one of the poles and the horizontally aligned smaller upper
ellipse contains the other pole. There is also a small region of overlap in the middle,
where poles are concentrated.
correspond to bound and virtual states; but we will not attempt a detailed analysis.
5
Conclusions
Our motivation for P-matrix parameterization was that it reflected an important
physical property about mesons, the quark confinement. For the first part of our
analysis, we tried to analyze the scattering data through P. We were able to achieve
statistically successful fits (X2 - 1) with very few parameters (3 for the cases we
examined).
It was clear in the I = 2 r7rscattering data that a blind fit may result in unwanted
behavior of the phase shift curve outside the range of our data. Nevertheless, our
P-Matrix Theory
21
fitt:ing parameters had small uncertainties and the fit inside the data region was
successful.
The next step was to generalize our results to complex k values by constructing
S, so that we could examine significantpoles. This part of our method had subtleties
to analyze.
We showed on an example that S we constructed is robust under the changes of
the parameters of P parameters. However,starting from a compensation P that has
infinitely many poles gives rise to an infinite array of poles in S, which seems to be
a generic case for our method. In fact, this can be confirmed by modifying Po by
keeping only finitely many poles of it, and then making the fits with this modified P
to construct S. Some preliminary analysis has showed that S had the same number
of poles
of poles
ignored.
artifacts
the first
For I =
as our beginning modified P0. This strongly suggests that the infinite array
is a mathematical artifact due to the infinite poles in Po, and can safely be
The delicate issue is, however, it is not always clear which poles are the
and which correspond to genuine physical resonances. For the I = 0 case,
two off-axes poles nearest to the origin are worth considering for future study.
2 case, again there is a pole significantly nearer to the real k-axis compared
to others, but it does not seem totally safe to attribute physical significance to this
pole.
In light of what we said, the problem of distinguishing the significant poles from
artifacts might be resolved by keeping finitely many poles of Po and taking this
modified form as the starting point of our fitting scheme, instead of P0 . This slightly
undermines our previous idea of Po being a natural starting point to parameterize P.
Nevertheless, P0 still has significance since we are still guided by the residues and the
locations of its poles, to have a rough idea of where we would have the parameters
of P. The main further direction is using this alternative method and trying to see
if the poles that seem to be physically significant in our current method are still
distinguished from the others.
Overall, P-matrix is an important method demonstrating the virtues of considering the physical system under consideration while analyzing scattering data. In cer-
tain cases, I = 0 s-wave 7rirscattering for example, it may lead to interesting results
about resonances, while in some other cases the relation of the results to underlying
physics is less clear. We hope further investigation of P will clarify the meaning of
the poles of S further and give more insight to the physics of quark systems.
22
F.M Ramazanoglu
Acknowledgments
The author is grateful to R. L. Jaffe for his help and supervision during all the stages
of the research that led to this work, and to M. R. Pennington for informing us of
the references for the scattering data.
References
[1] J.J. Sakurai, Modern Quantum Mechanics, Addison-Wesley Longman (1998)
[2] R.L. Jaffe, F. Low, Phys Rev D 19 (1979) 2105
[3] E.P. Wigner, L. Eisenbud, Phys. Rev. 72 (1947) 29
[4] H.C. Ohanian, Principles of Quantum Mechanics, Prentice Hall (1990)
[5] L. Liboff, Introductory Quantum Mechanics, Addison-Wesley (2003)
[6] R.L. Jaffe, Asyptotic Realms of Physics. Essays in Honor of Francis E. Low, MIT
Press (1983)
[7] M.R. Pennington, Personal comunication.
[8] W. Hoogland, et al., Nucl. Phys. B 126 (1977) 109
[9] S. Pislak, et al., Phys. Rev. Lett. 87 (2001) 221801
[10] S. Aoki, et al., Nucl. Phys. Proc. Suppl. 106 (2002) 230-232
[11] R.L. Jaffe, Hand Written Notes on P-Matrix (2005)
P-Matrix Theory
A
23
P-matrix for higher angular momentum waves
Assume we have a spherically symmetric finite-range potential in three dimensions
with dimension b. For this case we can treat the radial problem as a one dimensional
one by using the partial wave analysis. In this part I follow Ref.[1]. The basic
discussion of the R-Matrix (although its name is not mentioned) is in chapter 7.6.
A.1
Basic Partial Wave Analysis
I assume familiarity with the basics and will state the results from Ref.[1] without
detail.
The basic equation is
00oo
f(k, k')= kf(0) = Z(21 + 1)fi(k)I(cos0)
1=0
In the large r limit
(x (+) )
1
F.
~(2r)3/2
[ekz
+ f()
eikr
r
(21)
=
(2r)/2E(21+1)2ik[1 +2ik]--
)
(22)
where we used
ek
=
(2) 3/2 Z(21 + 1)i'j (kr)P (k
. r)
and the large r behavior of jl.
By unitarity, we can write
1 + 2ikfi(k) = e2i
.
The real number dt contains the information we need about the channel with angular
momentum I and
(2)3/2
A.2
21 + )P
e2i ' e
P-Matrix in 3 Dimensions
In this section we mainly follow Ref.[11] and Ref.[2].
-
( T
(23)
F.M Ramazanoglu
24
As we analyzed before, scattering in three dimensions can be analyzed by treating
each angular momentum value as a one dimensional problem. In this case, instead
of eikr, we use e, which are the radial wave functions of incoming and outgoing
definite momentum states. For example, e + = izh(l) and e = -izh(2) for angular
momentum 1. The coefficients are chosen so that e = eikr and eo = e - ik. Then we
can generalize our previous equation
V+ = el (kr) - Se+(kr)
p= +'(kb)
ke?'(kb)- e+'(kb)S
4p+(kb)
el (kb) - e+(kb) S
which lead to
Xe-(z) - el'(z)
-et(z) - et'(z)
Here, the scattered term has - sign since we are interested in the regular solution
and for the s-wave and S = 1
h( -(-)ho
= 2jo
+
sin
Z
which is finite at the origin.
For the free particle case, i.e. S = 1
k(krjl(kr))'
&,0=
A.3
krjl(kr)
I
kr=z
b
())
+
(z)
b dependence of the P-Matrix
Once we are outside the range of the potential (b), increasing the parameter b does not
change the physics, thus S. For this to happen, P should satisfy a certain relation.
d
+ _ ke+') = Pe- - ke-'}
{S(Pje
=~ S(Pike+'+ P,'e+ - k2e+")= Pke' + P,'e-- k2e- "
where all derivatives are with respect to z = kb. Assuming dS/db = 0 and using the
Schrodinger equation
d2
dr
2
l(
1(1+
+
r
=
k2e
2
k2e,,
[1( + 1)
] 2e
ke _ Clk2e
P-Matrix Theory
25
and the previous formula for S
(PeT - ke ) (Pke+' + Ple+- k2e+")= Pke' + Pl'e- - k2e-"
2 - P)e-)
(Pe-- ke') (Pike+'
- (Clk2 - P')e+)= (Pe+- ke+') (Pe-'- (Clk
k(pU+ P12- k 2 Ci)(e-e+'
-
e+e-') =
db = - P 2 - k2 [1- (+)]
Note that this is a simple generalization of our previous result in one dimension.
A.4
Data Analysis for Single Higher Angular Momentum
Channels
Using our result for the R, the compensation P for higher angular momentum waves
can be written as
P,o= b1+ jit('
Following the same logic of moving the poles to have the general P, we first note
that at a pole of P,, at , we need jl((n) = 0 and nearby the pole
(Z - Wn)J(C.)
Jl(z)
which gives a residue of ,n/b. So our general P is
1 (zj(z)
_
-
_-__2
1_
Z -
Z_
Z-Z2
Rearranging the previous equation to have P dependence explicit, we have
,
(
) e+(z)
e+'(z)
(n)z(-jL+ in,)- z(-j + in) - (-jl + in,)
()
z(ji + in,) - z(j + in') - (
(ij+zi[- zj
-(jr+zj[e2 i 81
e-iaLc
-
ei
)
+ in,)
-i (n + zn_-zn)
j ) -i(n +zn - z-n)
_i(-2a--7r)
26
F.M Ramazanoglu
where
(n,+ znI-
tan al
j)
This give the solution
-tan(Sl
B
+
2
) = (nr + z'-
!-no)
(j
1 + jI - "ZPj
Multichannel Scattering P-matrix
In real experiments, there are more than one possible outgoing channels and ingoing
channels. In this case, S and P become true matrices and the wave functions have
a component in each channel, thus become vectors. The first minor modification is
the normalization of the wave functions, so that their modulus does not depend on
Likewise our starting point for the one dimensional case, we first write down the
scattering states for an incoming wave in just a single channel
1
( )= e-(kjr)ij -
1
+
e+(kjr)Sji.
(24)
and define the matrix
ji= -e-(kjr)i - e+(kr)Sji= e----e+
The following functions will be essential in defining P:
s(kr) = K1 (e+(kr)e-(z) - e-(kr)e+(z)), s'(kb) = 1
c(kr) = K2 (e+(kr)e-'(z) - e-(kr)e+'(z)) , c(kb)= 1.
Note that, at r = b, s and r-derivative of c vanish.
Let us define the following states
Pji -
cij
c 6ijc(kjr) + ¾s(kjr).
We assume that these states can be written as a linear combination of the Scattering
States and these can be matched at r = b, so:
ij=
Ei Ail, 1j
c(kjr) +
ij
Vk--j
k
kj1kj
s(kjr) =
1i (ie-(kjr) - Slje+(kjr)).
27
P-l'atrix Theory
Again, by equating the two sides and their derivative at r = b
= yAi, (6ije-(zj)- Sje+(zj))
1..ji
R =A(e- - Se-
S = (eA
/~
Pj i
-e+)-
-
Ail (lje-'(zj) - lje+'(zj))
=
P = A(e-' - Se+')/k
- e+)-(e
= V(e-
-
- '
-
-e+')k
( $ = (e v - ( -e-')(e+
- e+)-l
For the derivative w.r.t b
db
=_ db
{/-( e - - e+)-1(e- - Se+')/-}
db
db
=
/-(e- - Se+)-l(e- '- e+')k(e- - Se+)-l(e- ' - Se+')/k
+\/k(e- - e+ )- (e- - e+")kVk=
-P2+ k2
where C is the diagonal matrix with entries C and where we used the matrix identities
(As)' = A +
', (A-')'= -A-A'A- 1.
For multichannel s-wave scattering, Co = -1, we reach exactly the same formula as
the single channel case:
Ip = _2 _ k2
Near a pole, assume we can write
=
U
k -ko
which leads to
U
dko
(k- ko)2 db
dUJ 1
_
U2
(k - ko)2
db k - ko
=> U2= _udk
d2=
=
Q 2 =Q
F.M Ramazanoglu
28
where
Q=-U
dko)-1
which is a projection matrix.
The explicit form of the s-wave scattering matrix can be written using e± = e±ikr
e
= eikr
S = e-ikb( + i)( - i)-le - ikb
=
S = -e-ikb(l - iIP)(1+ iP)-le - ikb
where
1
-
1
The easiest case to write the S explicitly is the 2-channel case:
(1
1 ip)-
-1
1 + iP1
iPl2
1
+iP21
1 + iP2 2
1 + iTrP+ detP
(
1
e -ikb
1+ iTrP - detP
e-ikb
_
_
1- i
-iP
1
1
-iP21
) (
( 1 + i(P22 - P11) detP
1+ iTrP - det
-iP21
1 + iP2 2
-iP12
1-iP22
-iP12
(
-2iP 21
1+i1 )
-iP12 ) e-ikb
1+ ilJ
-2iP1 2
1+ i(P1 - P 22 ) + detP
e
e-ikb
Trace and determinant of P and P are related as
TrP= P + P22
kl
k27
detP
detP = detk -
1
det.
ki kdetk
2
When one of these channels are closed, for example due to lack of energy to reach
the reaction of the channel, the wave number k becomes imaginary
_
kl --* k,
k 2 - i.
In this case, our scattering matrix is reduced to S1l
1+ i ( _
) +
+ i (P + k)
1+
+ i (
1 - iP k' e-2ikb
1 +i
Pd
kc
2detP
idetP
+
)
P-lMatrix Theory
29
which has exactly the same form as the 1-channel s-wave S-matrix. The reduced P,
Pred iS
P
p +detP
+14P22
:
P12P21
p1122
~+P22
K