Effective Field Theory

vergence structure. The NNLO for the QCD case is in [22] and the divergence structure in [23]. The Lagrangian constructed in [22] is with the changes dis-cussed in [1] and in Sect. II.2 also a complete Lagrangian for the other two cases but we have not shown it to be minimal nor calculated the divergence structure.

We do not repeat the discussion of the three different cases at the underly-ing fermion (quark) level. This can be found in [15] and [1], Sect. 2. In Sect.II.2 we quote the structure of the effective field theories for the three cases but we again refer to [1] for more details. Sect. II.3 discusses in detail the gen-eral structure of the amplitude. The amplitude can be expressed in terms of two functionsB(s, t, u)andC(s, t, u)which are generalizations of the amplitude A(s, t, u) inππ-scattering [24]. We work out the possible intermediate states using the relevant group theory and using a projection operator formalism obtain the amplitudes in the different channels. The results for the ampli-tude are discussed in Sect.II.4 and for the scattering lengths in Sect.II.5. Here we present some representative numerical results for the scattering lengths as well as some large n relations between the different cases. The length-ier formulas at two-loop order are given in an appendix. This work needed a few more integrals at intermediate stages than [25, 26], these are given in App.III.1. In Sect.II.6 we summarize our results.

II.2 Effective Field Theory

II.2.1 Generators

The notation for the three cases can be brought in a very similar form. More details can be found in [1]. The Goldstone boson live on a manifold G/H where G is the full global symmetry group and H is the part that remains unbroken after spontaneous symmetry-breaking. We label the unbroken gen-erators asT^aand the broken ones asX^a.

The spaceSU(n) ×SU(n)/SU(n)is isomorphic toSU(n)so we use theX^aas the generators ofSU(n)for the QCD case. They are traceless, hermitiann×n matrices.

The adjoint or real case has the generators inSU(2n)/SO(2n)where the bro-ken generators satisfy

J_SX^a= (X^a)^TJ_S, with J_S= ^{0 I} I 0

!

. (II.1)

Iis then×nunit matrix and the superscriptTindicates the transpose. TheX^a are traceless, hermitian2n×2nmatrices in this case. Multiplying (II.1) withJ_S

II

68 Meson-meson Scattering in QCD-like Theories

from left and right leads immediately to

X^aJ_S=J_S(X^a)^T . (II.2) The two-colour or pseudo-real case has the generators inSU(2n)/Sp(2n) where the broken generators satisfy

J_AX^a= (X^a)^TJ_A, with J_A= ⁰ −^I I 0

!

. (II.3)

TheX^aare traceless, hermitian2n×2nmatrices also in this case. Multiplying (II.3) withJ_Asimilar to above gives

X^aJ_A=J_A(X^a)^T . (II.4) The unbroken generators satisfy

SO(2n): T^aJ_S+J_ST^aT =0 ,

Sp(2n): T^aJ_S+J_ST^aT =0 . (II.5) This allows in both cases to derive usingJ=J_SorJ=J_Arespectively:

h^†J=Jh^T with h=exp ih^aT². (II.6) We always use generators normalized to one:

h^TaTbi = h^XaXbi =^δab. (II.7) hAi = tr_F(A), is the trace over the flavour indices. This is overnfor the QCD case and2nfor the real and pseudo-real case.

During the course of the calculation, we often have to sum over the Gold-stone Bosons. These sums can be easily performed using

complex :

tr_F(X^aAX^aB) = tr_F(A)tr_F(B)−¹

ntr_F(AB) , trF(X^aA)trF(X^aB) = trF(AB)−¹

ntrF(A)trF(B) . Real :

tr_F(X^aAX^aB) = ¹

2tr_F(A)tr_F(B) +¹ 2tr_F

AJ_SB^TJ_S

− ¹

2ntr_F(AB), tr_F(X^aA)tr_F(X^aB) = ¹

2tr_F(AB) +¹ 2tr_F

AJ_SB^TJ_S

−_2n^1trF(A)tr_F(B). Pseudoreal :

tr_F(X^aAX^aB) = ¹

2tr_F(A)tr_F(B) +¹ 2tr_F

AJ_AB^TJ_A

− ¹

2ntr_F(AB), tr_F(X^aA)tr_F(X^aB) = ¹

2tr_F(AB)−¹ 2tr_F

AJ_AB^TJ_A

− ¹

2ntrF(A)trF(B). (II.8)

II.2Effective Field Theory 69

There is a relation that the broken generators satisfy for the real and pseudo-real case.

tr_F

X^aX^b. . . X^kX^l

=tr_F

X^lX^k. . . X^bX^a

. (II.9)

The proof for the real case is tr_F

X^aX^b. . . X^kX^l

= tr_F

X^aX^b. . . X^kX^lJ_S²

= tr_F

X^aX^b. . . X^kJ_SX^lTJ_S

= tr_F

X^aX^b. . . J_SX^kTX^lTJ_S

= trF

J_SX^aTX^bT. . . X^kTX^lTJ_S

= trF



X^aTX^bT. . . X^kTX^lT

= tr_F

X^lX^k. . . X^bX^a_T

= tr_F

X^lX^k. . . X^bX^a

(II.10) The pseudo-real case is proven by replacingJ²_Sby −J²_Aand following the same steps. (II.9) is also the reason why the Lagrangian in [22] is not minimal for the real and pseudo-real case.

In the group theory references there is a conjecture mentioned that to get fromSO(2n) toSp(2n)it is sufficient to take n → −ⁿ. This feature is indeed visible in most of our formulas.

II.2.2 Lagrangians

As described in more detail in [1] we can write the Lagrangians in the three cases in a very similar way. The Goldstone Boson manifold G/H is parametrized by

u=exp

 i

√2Fφ



, φ=φ^aX^a. (II.11) These transform under the symmetry transformation in the QCD case for g_L×^gR∈^SU(n)_L×^SU(n)_Ras

u→gRuh(gL, gR, φ)^†=h(gL, gR, φ)ug^†_L. (II.12) his the socalled compensator field and is defined by (II.12) and is also anSU(n) matrix. This can be derived from the standard general formulation [27, 28] as done in [1]. For a transformation in the conserved part of the group we have thatg_L=g_R=g_Vandh=g_V.

II

70 Meson-meson Scattering in QCD-like Theories

The notation for the other two cases is directly that of [27, 28]. A symmetry transformationg∈G=SU(2n)transformsuas

u→g u h(g, φ)^†, with h=exp(ih^aT^a). (II.13) I.e.his in the unbroken partHof the group. In case the transformationgis in the conserved part of the group,g∈^H, we have thath=g.

We can now define the quantities u_µ = i

u^†∂_µu−^u∂µu^† , Γ_µ = ¹

2



u^†∂µu−u∂µu^†

. (II.14)

Under the group transformation in all cases we haveu_µ →^huµh^† and^Γ_µcan be used to define a covariant derivative.

∇µuν≡∂µuν+Γ_µuν−uνΓ_µ→h∇µuνh^†. (II.15) In [1] we also showed how the external fields can be included in a similar way as for the QCD case in [19, 29]. In particular the quark masses can be put in a quantityχ_±that transforms asχ_± →hχ_±h^†.

The lowest order Lagrangian takes on the standard form LLO= ^F2

4 tr_F u_µu^µ+χ+

(II.16) for all three cases and the same is true for the NLO Lagrangian.

LNLO = L0hu^µu^νuµuνi +L₁hu^µuµihu^νuνi +L2hu^µu^νihuµuνi +L₃hu^µuµu^νuνi +L₄hu^µuµihχ+i +L₅hu^µuµχ+i +L₆hχ+i² +L₇h^χ−i²+¹

2L₈h^χ2++χ²₋i. (II.17) We have kept only the terms contributing to meson-meson scattering in (II.17).

The NNLO Lagrangian is known for the complex or QCD case [22] as well as its divergence structure [23]. The same Lagrangian with the changes men-tioned above is complete for the other two cases but probably not minimal.

We have nonetheless chosen to leave the contributions from those terms in the results quoted here.

II.2.3 Renormalization

We use the standard renormalization procedure in ChPT [19, 29] with the ex-tension to NNLO described in great detail in [23, 26]. The divergences at NLO are canceled by the subtractions as calculated in [1]. At NNLO the divergences