Redundancy of the off-shell parameter Z

The description of the interacting spin-3/2 fields leads to the introduction of an addi-tional off-shell parameter for each spin-3/2 field. The off-shell parameter describes to which amount the lower-spin component of the field operator contribute to the physical observables (see e.g. [69]). In the framework of the effective field theory these off-shell parameters can be absorbed, independent of whether they can be fixed by theoretical arguments or by fitting to the experemental data or not, by the contact interaction terms which are of higher order in the energy expansion [86, 58]. This redundance can also be understood in terms of a suitable field redefinition, which eliminates the contribution of the lower-spin component in the original interaction at the cost of the introduction of additional interaction terms [74].

In the following we consider a derivative coupling of a scalar spin-0 field φ to a spin-1/2 field ψ and a spin-3/2 Rarita-Schwinger field ψµ with a minimal number of derivatives.

The associated masses are m, M and M³

2. We apply the ideas of Tang and Ellis in [86]

and generalize them to theSU(3) chiral version of this vertex and to the case of arbitrary number of dimensions d¹.

Starting with the tree-point vertex

L=C ψ¯_µΘ^µνψ ∂_νφ+ h.c.

, Θ_µν =g_µν −1

2Zγ_µγ_ν, (A.16) (C is the dimensionful coupling, Z is the off-shell parameter) the s- and u-channel di-agrams with the exchange of the spin-3/2 particle can be obtained by means of the following effective four-point interaction:

L^{ef f} =−C²ψΘ¯ _µλS^λσΘ_σνψ ∂^µφ ∂^νφ, (A.17) with

S_µν = −1 i/∂−M³

2 +iε

g_µν − 1

d−1γ_µγ_ν− i (d−1)M³

2

(γ_µ∂_ν −γ_ν∂_µ)− d−2 (d−1)M²3

2

∂_µ∂_ν

. (A.18) The spin-3/2 propagator in the coordinate space

iS_µν(x−y) =

Z d⁴k

(2π)⁴iS_µνe^ik(x−y) (A.19)

satisfies the equation

Λ^µσS_σν(x−y) =−δ⁽⁴⁾(x−y)g^µ_ν, (A.20) Λµν being the differential operator that determines the kinetic term of the spin-3/2 field in the Lagrangian:

Λ_µν =− i/∂−M³

2

g_µν +i(γ_µ∂_ν+γ_ν∂_µ)−γ_µ i/∂+M³

2

γ_ν. (A.21) Using the property

γ^µS_µν =S_νµγ^µ= 1 (d−1)M³

2

d−2 M³

2

i∂_ν −γ_ν

!

, (A.22)

and

Θ_µλS^λσΘ_σν = S_µν− Z²(d−2) 4(d−1)M²3

2

(g_µν−iσ_µν)i/∂ − Z(Zd−4) 4(d−1)M³

2

(g_µν −iσ_µν) + Z²(d−2)

2(d−1)M²3 2

γ_µi∂_ν − Z(d−2) 2(d−1)M²3

2

(i∂_νγ_µ+γ_νi∂_µ), (A.23)

1The analysis can be carried out for other vertices linear in spin-3/2 fields

the effective Lagrangian in (A.17) can now be written as L^{ef f} = −C²ψS¯ ^µνψ ∂_µφ∂_νφ

+ R^(S,T)

ψψ ∂¯ ^µφ ∂_µφ−ψiσ¯ ^µνψ ∂_µφ ∂_νφ + R^{(S,/ T/)}

ψi/¯ ∂ψ ∂^µφ ∂_µφ−ψiσ¯ ^µνi/∂ψ ∂_µφ ∂_νφ

+ 1

2R^(V)

ψγ¯ _µi∂_νψ ∂^µφ ∂^νφ+ h.c.

= −C²ψS¯ ^µνψ ∂_µφ∂_νφ + h

R^(S,T)+M R^{(S,/ T/)}i

ψψ ∂¯ ^µφ ∂_µφ−ψiσ¯ ^µνψ ∂_µφ ∂_νφ

+ 1

2R^(V)

ψγ¯ _µi∂_νψ ∂^µφ ∂^νφ+ h.c.

+ R^{(S,/ T/)}

ψ(i/¯ ∂−M)ψ ∂^µφ ∂µφ−ψiσ¯ ^µν(i/∂−M)ψ ∂µφ ∂νφ

, (A.24) with

R^(S,T) =C² Z(Zd−4) 4 (d−1)M³

2

, R^(/S,T/)=C² Z²(d−2) 4 (d−1)M²3

2

, R^(V)=−C²Z(Z−2) (d−2)

2 (d−1)M3² 2

. (A.25)

The Z-dependent terms in (A.24) can be absorbed by the Q² interaction terms of the form

L⁽²⁾ = g^(S)ψψ ∂¯ ^µφ ∂µφ+g^(S/)ψi/¯∂ψ ∂^µφ ∂µφ

+ g^(T)ψiσ¯ ^µνψ ∂_µφ ∂_νφ+g^(T/)ψiσ¯ ^µνi/∂ψ ∂_µφ ∂_νφ + 1

2g^(V)

ψγ¯ _µi∂_νψ ∂^µφ ∂^νφ+ h.c.

= h

g^(S)+M g^(/S)i

ψψ ∂¯ ^µφ ∂_µφ+h

g^(T)+M g^(T/)i

ψiσ¯ ^µνψ ∂_µφ ∂_νφ + 1

2g^(V)

ψγ¯ _µi∂_νψ ∂^µφ ∂^νφ+ h.c.

+ g^(/S)ψ¯(i/∂ −M)ψ ∂^µφ ∂_µφ+g^(T/)ψiσ¯ ^µν(i/∂ −M)ψ ∂_µφ ∂_νφ. (A.26) Upon the redefinition

g^(S) →g^(S)+M g^(S/), g^(T)→g^(T)+M g^(T/), (A.27) the absorption of the dependence on the off-shell parameter Z in the vertex (A.16) amounts to a finite renormalisation of the Q² couplings in the above Lagrangian ac-cording to

δg^(S) = R^(S,T)+M R^{(S,/ T/)}, δg^(V) = R^(V),

δg^(T) = −

R^(S,T)+M R^{(S,/ T/)}

. (A.28)

At one loop-level, the Z-dependent part in the contribution to the self-energy of the spin-1/2 particle, Σ^Z, calculated with (A.16), is Q⁴ and can be fully absorbed into the contributions from the Q⁴ tadpole diagrams, Σ⁽⁴⁾, calculated with (A.26). The explicit form of the both contributions in dimensional regularisation is

Σ^Z(p) =

Z d^dk (2π)^d

iµ^4−d_{U V} k²−m²+iε

h

C²k_µΘ^µλS_λσ(p−k)Θ^σνk_ν−C²k_µS^µν(p−k)k_νi

= −

Z d^dk (2π)^d

iµ^4−d_{U V} k²−m²+iε

h

R^(S)k²+R^(S/)/p k²+R^(V)/k(k·p)i

= −

Z d^dk (2π)^d

iµ^4−d_{U V} k²−m²+iε

h

R^(S)+/p

R^(S/)+ 1 dR^(V)

i m²,

= −

Z d^dk (2π)^d

iµ^4−d_{U V} k²−m²+iε

×h

R^(S)+M R^(S/)+1

dM R^(V)

+ (/p−M)

R^(S/)+ 1 dR^(V)

i m²,

(A.29) Σ⁽⁴⁾(p) = −

Z d^dk (2π)^d

iµ^4−d_{U V} k²−m²+iε

h

g^(S)k²+g^(S/)(/p−M)k²+g^(V)/k(k·p) i

= −

Z d^dk (2π)^d

iµ^4−d_{U V} k²−m²+iε

h

g^(S)+g^(/S)(/p−M) +/p1 dg^(V)i

m²

= −

Z d^dk (2π)^d

iµ^4−d_{U V} k²−m²+iε

h

g^(S)+1

dM g^(V)

+ (/p−M)

g^(S/)+ 1 dg^(V)i

m². (A.30)

At one-loop level we observe that the expressions (A.29) and (A.30) preclude one from drawing the conclusions about the renormalisation of each coupling constant in (A.26) separately. However, it is possible, as expected, to absorb all the Z-dependence into the tadpole contributions in Σ⁽⁴⁾ in complience with the results in (A.28).

The results in (A.24) and (A.28) are generalised to the SU(3) invariant interaction by mapping the flavour structure in theSU(3) version of (A.24) to the flavour basis defining the flavour structures of the Q² terms stated in Section 1.4. It holds:

δg₀^(S) = 4 3

R^(S,T)+R^{(S,/ T/)}

, δg₁^(S) =−1 3

R^(S,T)+R^{(S,/ T/)}

, δg_D^(S) = −

R^(S,T)+R^(/S,T/)

, δg_F^(S) = R^(S,T)+R^{(S,/ T/)}, (A.31)

δg₀^(V) = 4

3R^(V), δg₁^(V)=−1 3R^(V),

δg_D^(V) = −R^(V), δg_F^(V) = R^(V), (A.32)

δg^(T₁ ⁾=R^(S,T)+R^{(S,/ T/)}, δg_D^(T) =−

R^(S,T)+R^(/S,T/) , δg_F^(T) =−1

3

R^(S,T)+R^{(S,/ T/)}

. (A.33)

Appendix B.

SU (3) group theory

The importance of the SU(3) Lie-group in physics is subject of many textbooks (see e.g. [33, 59, 44]). In this chapter the main properties of the generators of the group transformations are summarised¹. Furthermore, the decomposition of the product of two arbitrary SU(3) octets is discussed.

The main properties of the generators of the SU(3) group t^a= 1

2λ^a, (a= 1, . . .8), (B.1)

λ^a being the Gell-Mann matrices

λ¹ =





0 1 0 1 0 0 0 0 0



, λ² =





0 −i 0 i 0 0

0 0 0



, λ³ =





1 0 0

0 −1 0

0 0 0



,

λ⁴ =





0 0 1 0 0 0 1 0 0



, λ⁵ =





0 0 −i 0 0 0

i 0 0



, λ⁶ =





0 0 0 0 0 1 0 1 0



,

λ⁷ =





0 0 0 0 0 −i 0 i 0



, λ⁸ = 1

√3





1 0 0

0 1 0

0 0 −2



, (B.2)

are

t^a† =t^a, tr(t^a) = 0, tr(t^at^b) = 1 2δ^ab, [t^a, t^b] =if^abct^c, {t^a, t^b}= 1

3δ^ab+d^abct^c. (B.3) From this, one obtains for the product of two generators

t^at^b = 1

6δ^ab+ 1

2 d^abc+if^abc

t^c, (B.4)

1For further relations we refer to the mentioned textbooks or to [13].

and thed- andf-symbols of the SU(3) group are given by d^abc = 2 tr t^a{t^b, t^c}

, f^abc=−2itr t^a[t^b, t^c]

. (B.5)

In general, tensors or products of them are reducible in the sence that they can be further decomposed into components, which transforms under different irreducible rep-resentations of the symmetry group. Irreducible reprep-resentations can be specified by the dimension of the multiplet - orthonormal basis of the support of the representation.

Working with the effective quark operators in Sections 2.3.1 and 2.3.2 it is more appropri-ate to use SU(3) flavour octets described by a single adjoint index a= 1, . . .8 instead of the indicesi, j = 1,2,3 that are used to label vectors transforming under the fundamental representation. Given a general octet tensor O^j_i, the connection is established via:

O^a= (t^a)ⁱ_jO^j_i, and Oⁱ_j = 2O^a(t^a)ⁱ_j. (B.6) In the following, the decomposition of the product of two arbitrary octets, Aⁱ_j and Bⁱ_j, is discussed. It holds:

8⊗8=1⊕8S⊕8A⊕10⊕10⊕27. (B.7) The singlet, 1, is obtained by contracting all indices:

S=Aⁱ_lB^l_i = 4A^aB^btr(t^at^b) = 2A^aB^a. (B.8) The symmetric octet, 8_S, in the product of two octets:

Dⁱ_j = Aⁱ_kB^k_j +Bⁱ_kA^k_j −2

3δⁱ_jA^k_lB^l_k= 4A^aB^b{t^a, t^b}ⁱ_j −2

3δⁱ_j4A^aB^btr(t^at^b)

= 4A^aB^b 1

3δ^ab+d^abct^ci j −2

3δⁱ_j4A^aB^b 1

2δ^ab = 4A^aB^bd^abc(t^c)ⁱ_j. (B.9) The symmetric octet, described by a single adjoint index, is given by

D^c= (t^c)ⁱ_jD^j_i = 2d^abcA^aB^b =d^abc(A^aB^b +A^bB^a). (B.10) Similar for the antisymmetric octet, 8_A:

Fⁱ_j = Aⁱ_kB^k_j−Bⁱ_kA^k_j = 4A^aB^b[t^a, t^b]ⁱ_j = 4A^aB^bif^abc(t^c)ⁱ_j, (B.11) and

F^c = (t^c)ⁱ_jF^j_i = 2if^abcA^aB^b =if^abc(A^aB^b −A^bB^a). (B.12) Decuplet and antidecuplet tensors in the product of two octets are:

T^ijk =Aⁱ_lB^j_m^klm+perm(ijk), T_ijk =A^l_iB^m_{j klm}+perm(ijk). (B.13) Another way to represent decuplet and anti-decuplet tensors uses the fact, that the totaly antisymmetric tensor of SU(3), ε^ijk =εijk, carries three indices. A pair of upper indices

in which the tensor is antisymmetric can be always traded for one lower index using an epsilon-tensor and similarly for the lower indices. It holds:

T_[lm]^(ij) = Aⁱ_lB^j_m+A^j_lBⁱ_m−Aⁱ_mB^j_l −A^j_mBⁱ_l

− 1

3 δⁱ_lF^j_m−δ_mⁱ F^j_l +δ_l^jFⁱ_m−δ^j_mFⁱ_l , T_(lm)^[ij] = Aⁱ_lB^j_m+Aⁱ_mB^j_l−A^j_lBⁱ_m−A^j_mBⁱ_l

− 1

3 −δ_lⁱF^j_m−δ_mⁱ F^j_l +δ_l^jFⁱ_m+δ_m^j F^j_l

. (B.14)

Here, the round and square brackets on T indicate the symmetrisation and the antisym-metrisation of the indices, respectively, and the subtraction of F’s, defined in (B.11), is required to make the tensors traceless. Converting the sum of these multiplets,10+10,¯

T_[lm]^(ij)+T_(lm)^[ij] = 2

Aⁱ_lB^j_m−A^j_mBⁱ_l− 1

3 δ_l^jFⁱ_m−δⁱ_mF^j_l

, (B.15)

to an object with two adjoint indices yields upon a short calculation:

T^ab = (t^a)^l_i(t^b)^m_j

T_[lm]^(ij)+T_(ij)^[lm]

= 2

A^aB^b−A^bB^a− 2

3f^abcf^cdeA^dB^e

. (B.16) Finally,27 inAⁱ_lB^j_m is obtained by symmetrizing upper and lower indices and by making this tensor traceless:

I_(lm)^(ij) = Aⁱ_lB^j_m+A^j_lBⁱ_m+Aⁱ_mB^j_l +A^j_mBⁱ_l

− 1

5 δ_lⁱD^j_m+δ^j_lDⁱ_m+δ_mⁱ D^j_l+δ_m^j Dⁱ_l

−1

6 δ_lⁱδ_m^j +δ^j_lδ_mⁱ

S. (B.17) Here, the subtracted terms contain the symmetric octet and singlet tensors, defined in (B.9) and (B.8), respectively. The 27-plet, described by two adjoint indices, is given by

I^ab= (t^a)^l_j(t^b)^m_j I_(lm)^(ij) = 2

A^aB^b+A^bB^a− 6

5d^abcd^cghA^gB^h− 1

4δ^abA^cB^c

. (B.18)

• • •

We summarise the results of this section in a form, which is well suited for the dis-cussion of the identities for the effective quark operators in Appendix G.

In general, a tensor with two adjoint flavour indices, χ^ab, can be decomposed into sym-metric and antisymsym-metric parts:

χ^ab = 1

2 χ^ab₊ +χ^ab₋

. (B.19)

Further decomposition of these parts into the different multiplets is obtained by using

the results in (B.8, B.10, B.12, B.16, B.18):

χ^ab₊ = 1

8δ^abχ^cc₊ +3

5d^abcd^cghχ^gh₊ + χ^ab₊ − 1

8δ^abχ^cc₊− 3

5d^abcd^cghχ^gh₊

= 1

8δ^abχ₁+ 3 5d^abcχ^c₈

S +χ^ab₂₇, χ^ab₋ = 1

3f^abcf^cghχ^gh₋ + χ^ab₋ −1

3f^abcf^cghχ^gh₋

= 1

3f^abcχ^c₈

A +χ^ab10+10¯ , (B.20)

with the symmetric components

χ1 =χ^cc₊, χ^c_8S =d^abcχ^ab₊, χ^ab₂₇=χ^ab₊ −1

8δ^abχ^cc₊ − 3

5d^abcd^cghχ^gh₊, (B.21) and with the antisymmetric components

χ^c₈

A =f^abcχ^ab₋, χ^ab_10+10¯ =χ^ab₋ − 1

3f^abcf^cghχ^gh₋. (B.22)

Appendix C.

Interaction

When dealing with effective field theories, where the degrees of freedom are described by SU(3) flavour tensors¹, one is confronted with the problem of finding the minimal number of linearly independent SU(3)-invariant terms. This number is obtained by counting the singlets in the decomposition of the outer product of the SU(3) tensors under consideration. To determine the terms themselves, a general method for octets only was formulated in [24].

The first section of this appendix presents group theoretical methods, similar in spirit to [24], for the construction of flavour structures that are needed to fully describe the Q² four-point meson-baryon interaction which is stated in Section 1.4.2. Results obtained for this vertex are then applied with small modifications to the construction of Q⁴-terms in Section 1.4.3 that break the chiral symmetry explicitly. The analysis is carried out for octet and decuplet baryons.

The second section discusses transformation properties under charge conjugation of the building blocks of the four-point meson-baryon interaction.

Im Dokument On the quark-mass dependence of baryon ground-state masses. (Seite 86-95)