Abstract. New unitary symmetry relations are obtained between the QCD sum rules for ΣΛ transitions. Also for QCD SR&apos;s with baryon distribution amplitudes the unitary relations are constructed, proved in a simple quark model and found to be consistent with the existing calculations

Altug Ozpineci

Valeri Zamiralov

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New Unitary Relations between the QCD sum rules
Altug Ozpineci1,a and Valeri Zamiralov2,b
1Physics Department, Middle East Technical University, 06531 Ankara, Turkey
2Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,119991 Moscow, Russia
Abstract. New unitary symmetry relations are obtained between the QCD sum rules
for ΣΛ transitions. Also for QCD SR’s with baryon distribution amplitudes the unitary
relations are constructed, proved in a simple quark model and found to be consistent with
the existing calculations.
1 Introduction
QCD sum rules (QCD SR’s) introduced by [1, 2] have become one of the most important tools while
studying baryon or meson couplings and the corresponding form factors with photons, mesons or
baryons. At the same time, the complexity of the formulas used in calculations has increased in
obviously non-linear manner. So the control of one’s own and other authors calculations becomes a
non-trivial problem. Recently, we have proposed simple relations between the quantities involving Σ
hyperons and those involving Λ hyperons [3]. They are generalized easily to any Σ-like and Λ -like
baryons containing heavy quarks. Our formulas seem to be useful and for example they were inserted
into the codes by Lai Wang and Frank Lee while studying magnetic moments of the octet baryons [4].
It is interesting that also the transition Σ − Λ quantities could be obtained from the corresponding
quantities of the Σ or Λ baryons. We want here to propose one more relation to control the complex
calculations of the transition Σ−Λ quantities which has the merit that it does not involve other baryon
quantities, neither of Σ nor of Λ baryons.
The formula is rather trivial for nonrelativistic quark model or simple unitary symmetry model.
However, it shows us the way to overcome many difficulties in the framework of QCD sum rules, as
we shall see later on. In the very tedious QCD sum rule calculations it proves to be extremely useful.
Its applicability was demonstrated in [3] on the example of sum rules for magnetic moments of octet
baryons. In this report we are interested with the Σ − Λ transitions. In [3] they were related to the Σ
and Λ quantities as (we write in terms of correlators [1, 2])
ΠΣ
0(d↔s) − ΠΣ0(u↔s) = −ΠΛ(d↔s) + ΠΛ(u↔s) = √3ΠΛΣ0 , (1)
where d ↔ s means interchange of the flavors d and s. The main point of our report is that it can
be formulated in some "auto-sufficient" way. Our new relation between the corresponding correlation
functions reads as
ΠΛΣ
0(u↔s) + ΠΛΣ
0(d↔s) = ΠΛΣ
0
. (2)
ae-mail: ozpineci@metu.edu.tr
be-mail: zamir@depni.sinp.msu.ru
  
 
 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits 
unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 
DOI: 10.1051/
C© Owned by the authors, published by EDP Sciences, 2014
/2014 0004
, 000 (2014)ConferencesEP  Web ofJ 80
80ep confj
45
5
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20148000045
2 Master formulas for the octet
As the main ingredient of the polarization operator is the matrix element of the T-production of the
interpolating currents, we now put our attention to it. The corresponding formulas are often named as
"master formulas"[4]. We begin with the master formula for the Σ0 [4], [7]:
Σ0 ≡ 〈0|T {ηΣ0 η̄Σ0 }|V〉 = (3)
= −εabcεde f {S adu γ5CS bes Cγ5S c fd + S add γ5CS bes Cγ5S c fu +
S adu Tr[γ5CS
be
s Cγ5S
c f
d ] + S
ad
d Tr[γ5CS
be
s Cγ5S
c f
u ] + ...
where V could be meson or photon. It is vacuum state for the mass sum rule (SR). S is the quark
propagator [1],[2]. We write for our purpose just the first line, as for us it is important that in every
line the quark flavors index is of the form
Σ0 = u s d + d s u .
Now, we construct Λ−quantity using the relation from [3]
Λ ≡ 〈0|T {ηΛη̄Λ}|V〉 = (4)
εabcεde f {2S adu γ5CS bed Cγ5S c fs + 2S ads γ5CS bed Cγ5S c fu +
2S ads γ5CS
be
u Cγ5S
c f
d + 2S
ad
d γ5CS
be
u Cγ5S
c f
s
−[S adu γ5CS bes Cγ5S c fd + S add γ5CS bes Cγ5S c fu ] + ...
and we retain for a moment only these terms to show explicitly the group-theoretical structure. In
fact, every three lines as to the quark flavors are of the form
Λ = 2 u d s + 2 s d u + 2 s u d + 2 d u s − u s d − d s u
in agreement with [3].
Transition magnetic moment "master formula" could be read from [3] as
√
3Σ0Λ ≡ √3〈0|T {ηΣ0 η̄Λ}|0〉 = (5)
εabcεde f
{
[S adu γ5CS
be
d Cγ5S
c f
s + S ads γ5CS
be
d Cγ5S
c f
u ]−
[S ads γ5CS
be
u Cγ5S
c f
d + S
ad
d γ5CS
be
u Cγ5S
c f
s ] + ...
[S adu Tr[γ5CS
be
d Cγ5S
c f
s ] + S ads Tr[γ5CS
be
d Cγ5S
c f
u ]]−
[S ads Tr[γ5CS
be
u Cγ5S
c f
d ] + S
ad
d Tr[γ5CS
be
u Cγ5S
c f
s ]] + ...
One can see that every two lines as to the quark flavors are of the form
u d s + s d u − s u d − d u s
Canceling identical terms in the full expression, we obtain
√
3Σ0Λ = εabcεde f
{
[S adu γ5CS
be
d Cγ5S
c f
s + S ads γ5CS
be
d Cγ5S
c f
u ]− (6)
EPJ Web of Conferences
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[S ads γ5CS
be
u Cγ5S
c f
d + S
ad
d γ5CS
be
u Cγ5S
c f
s ]+
[S adu Tr[γ5CS
be
d Cγ5S
c f
s ] − S add Tr[γ5CS beu Cγ5S c fs ]] + ...
New relations (we restrict ourselves with the first terms) yield
Σ0Λ|(d↔s) + Σ0Λ|(u↔s) = (7)
εde f
{
[S adu γ5CS
be
s Cγ5S
c f
d + S
ad
d γ5CS
be
s Cγ5S
c f
u ]−
[S add γ5CS
be
u Cγ5S
c f
s + S ads γ5CS
be
u Cγ5S
c f
d ]
}
+
εabcεde f
{
[S ads γ5CS
be
d Cγ5S
c f
u + S adu γ5CS
be
d Cγ5S
c f
s ]−
[S adu γ5CS
be
s Cγ5S
c f
d + S
ad
d γ5CS
be
s Cγ5S
c f
u ] = Σ
0Λ .
One can see that every two lines, as to the quark flavors, are of the form
( u s d + d s u − d u s − s u d)+
( s d u + u d s − u s d − d s u) =
u d s + s d u − s u d − d u s.
Let us give an example. In [5] a QCD SR for the magnetic transition moment was written, which
reads in terms of various condensates aq = −(2π)2〈q̄q〉, b = 〈g2sG2〉 and so on [5],[6] (for a moment
we maintain au, ad different, the same for other condensates and quark masses mu, md):
S R(Σ0Λ) = (eu − ed) M
6
4L4/9
+
L4/9
9M2
[esas(au − ad) + euas(au − 3ad) (8)
−edas(ad − 3au)] + M
2b
24L4/9
(eu − ed)
M2b
144L4/9
[[ln(
M2
Λ2
) − 1 − γEM] + 4[ln( M
2
Λ2
) − γEM − M
2
2Λ2
]](eu − ed)
+[
−χ
6L4/27
(M2 −
m2(s)0
8L4/9
) × [as(euχuau − edχdad) + esχsas(au − ad)]+
M2
L4/9
1
4
[2(euaumd − edadmu) − 2esas(mu − md)−
as(eumd − edmu) − es(aumd − admu) + 2(edadmu − euaumd)+
2ms(edad − euau) + ms(euad − edau) + es(admu − aumd)]
+
L4/9
36
(2κ − ξ)] × [as(eu(2κu − ξu)au − ed(2κd − ξd)ad) + es(2κs − ξs)as(au − ad)]
− M
4
4L28/27
[(euauχu − edadχd)ms + esasχs(mu − md)]+
1
18
[(euau(2κu − ξu) − edad(2κd − ξd)ms + esas(2κs − ξs)(mu − md)]M2−
M2
6
[(euauκu − edadκd)ms + esasκs(mu − md)]×
×[ln( M
2
Λ2
) − 1 − γEM]] = βΣβΛ
√
3μΣ0Λe−m̄
2/M2 (1 + AΣ0ΛM2) + ...
One can see that each term in this formula satisfies our new relation
〈ΣΛ〉|(u↔s) + 〈ΣΛ〉|(d↔s) = 〈ΣΛ〉.
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3 Baryon Distribution Amplitudes
Our group-theoretical relations for the product of interpolating currents do not work if we go to higher
Q2 (baryon electromagnetic form factors at high momentum transfer, form factors of lepton beta-
decay of the heavy baryons etc). QCD SR’s now are formulated for the product of baryon interpolating
current and the photon or weak boson one. In the ket bracket we have now not photon, meson or
weak boson, but a baryon. Correspondingly, one should work not with photon or boson distrubution
amplitudes, but with the baryon distribution amplitudes [8].
The resulting formulas are even more cumbersome than in the case of QCD SR’s based on the
product of the baryon interpolating currents, and the final formulas of several authors for the same
subject are different. That is why we decided to search for some group theoretical relations also for
this case. New group-theoretical relations for the SR’s with baryon distribution amplitudes read as
(i)
√
3〈0|ηΛηγ|Λ〉 = 2〈0|η(u↔s)Σ ηγ|Λ〉 + 〈0|ηΣηγ|Λ〉 (9)
(ii)
√
3〈0|ηΛηγ|Σ〉 = 2〈0|η(u↔s)Σ ηγ|Σ〉 + 〈0|ηΣηγ|Σ〉
(iii)
√
3〈0|ηΣηγ|Λ〉 = −2〈0|η(d↔s)Λ ηγ|Λ〉 + 〈0|ηΛηγ|Λ〉
(iv)
√
3〈0|ηΣηγ|Σ〉 = −2〈0|η(d↔s)Λ ηγ|Σ〉 + 〈0|ηΛηγ|Σ〉
We have 4 different matrix elements and try to write the solutions in the NRQM as
〈0|ηΛηγ|Λ〉 = aeu + bed + ces.
Putting it into (iii) we get
〈0|ηΛηγ|Λ〉 = a(eu − ed),
√
3〈0|ηΣηγ|Λ〉 = −a(eu + ed − 2es).
Similar calculations can be made for relations (ii) and (iv) wherefrom general solution would be
〈0|ηΣηγ|Σ〉 = ã(eu + ed − 2es),
√
3〈0|ηΛηγ|Σ〉 = 3ã(ed − eu).
Thus solutions for the matrix elements 〈0|ηΣηγ|Σ〉 and
√
3〈0|ηΣηγ|Λ〉 should be similar up to some
overall factor. Surprisingly enough we have seen that these relations are valid for QCD sum rules in
[9, 10]. As an example we take ΣΛ transition formula from Eq.(9) in [10] (we write only the 1st part
of it and put ms = 0 which does not affect our conclusions)
FΣΛ1 (Q
2) =
1
4
√
2λΣ0
em
2
Σ0
∫ 1
x0
dx(−ρ2(x)
x
+
ρ4(x)
M2x2
− ρ6(x)
2M4x3
)e−(
Q2(1−x)
M2 x
+
m2
Λ
x
M2
)
+ ...
where
ρ6(x) = 4m3Λx(m
2
Λ + Q
2)(1 + β)(euB6(x) + edB̃6(x) + 2esB̂6(x)); (10)
ρ2(x) = −2eumΛ{[(1 − β)(D2 +C2) − (1 + β)(B2 − B4)](x)
+x
∫
dx3[(1 − β)(A3 + 2V1 − 3V3) − (1 + β)(P1 + S 1 − 5T1 + 10T3)](x, 1 − x − x3, x3)}
+2edmΛ{[(1 − β)(D̃2 − C̃2) + (1 + β)(B̃2 − B̃4)](x)
+x
∫
dx1[(1 − β)(A3 − 2V1 + 3V3) − (1 + β)(P1 + S 1 + 5T1 − 10T3)](x1, x, 1 − x1 − x)}
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+4es{mΛ[(1 − β)Ĉ2 − (1 + β)(B̂2 − B4)](x)
+
∫
dx1{(1 − β)(mΛxV3 + (1 + β)mΛx(2T3 − T1]}(x1, 1 − x1 − x, x)}
and we omitted ρ4 as it is too lengthy. All the A, B, C, etc., are defined in [10].
If we change all the Λ symbols to Σ0 ones we just arrive at the formulas for FΣ
0
1 (Q
2) for the Σ0
form factor of [9] (see the Appendix of that reference):
FΣ
0
1 (Q
2) =
1
4
√
2λΣ0
em
2
Σ
∫ 1
x0
dx(−ρ2(x)
x
+
ρ4(x)
M2x2
− ρ6(x)
2M4x3
)e−(
Q2(1−x)
M2 x
+
m2
Σ
x
M2
)
+ ... ,
where ρ2,4,6 are the same as in (10) with the only change of mΛ to mΣ, which proves the validity of our
relation for the baryon DA’s.
4 Conclusions
1. We have obtained the new group-theoretical relation for the QCD SR and proved it for general
"master formulas" as well as for the QCD SR’s for ΣΛ magnetic transition moment in terms of
various condensates. This relation is valid also for the QCD sum rules on the light cone.
2. We have established new relations for the case of baryon DA’s and have shown its validity on the
example of the LC QCD SR for the electromagnetic form factors of Σ baryon and Σ − Λ transition.
3. These relations are not constraint to the octet representation of SU(3) but also can be written for
sextet and antitriplet that is for ΣQ and ΛQ and for pairs of heavy cascade hyperons ΞQ and Ξ
′
Q with
Q = c, b with the appropriate changes.
We thank the Organizers of the Workshop QCD 2014 for their kind hospitality.
References
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243 (2005)
[4] Lai Wang and Frank X. Lee, Phys. Rev. D78, 013003 (2008)
[5] Shi-lin Zhu, W.-Y.P. Hwang, and Ze-sen Yang, Phys. Rev. D57, 1524 (1998)
[6] Charles B. Chiu, J. Pasupathy, and S.J. Wilson, Phys. Rev. D33, 1961 (1986)
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New Unitary Relations between the QCD sum rules

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New Unitary Relations between the QCD sum rules

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