^{3}

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in the effective

Chiral perturbation theory (

For

In a previous paper [

In particular we could convert the computation of the mass gap in a periodic box by Niedermayer and Weiermann [

Although

In this paper we do not analyze explicit chiral symmetry breaking. In QCD the effect of including a small quark mass on the finite volume spectrum has been computed for

The dynamical fields are matrices

In this limit the leading-order effective Lagrangian is given by [

For

The four-derivative couplings in Eq. (^{1}

For

A proof of Eq. (

From these relations it follows that the results obtained for general

The

Writing

This leads to the identification [

These and the relations (

Here we work in a continuum volume ^{2}

For the perturbative expansion we parameterize ^{3}

The effective action

The total effective action, including also the four-derivative terms, has then a perturbative expansion of the form

Note that

The free two-point function is given by

The chemical potential

This gives an additional

Further, writing

The four-derivative operators

The

Note that for an observable

Now

Averaging over the zero modes, denoting

Next

For the averages we have

Note that ^{4}

Note that for pbc

Using Eq. (

This has a perturbative expansion

Expanding Eq. (

Now

Here

So we have at leading order

At next order

First^{5}

This two-loop function, the “massless sunset diagram”, is calculated in detail in Ref. [

Secondly

Next

Furthermore

Note that

For the averages we have

One can check that

Finally

Collecting the results together, the expansion of the susceptibility with DR is given by

For

We first recall some results obtained in Ref. [

The shape function

Below we switch to the conventional couplings

By convention [

Requiring cancellation of the

Due to these relations the terms ^{6}

Finally one has

For

For the O(

Our result (

In this section we will compute the mass gap of the 4D chiral

Here we will only give a brief description of the computation since it closely follows that for the O(

It follows that the mass gap

Since

Eq. (

It thus suffices to compute the coefficients ^{7}

The fields

The corresponding free two-point function is given by

Expanding

The interaction terms in the total action have the same form as in the previous section apart from the integration range, which is now

The computation now proceeds as in Ref. [

In the next order

Finally at third order we obtain

The term

The third-order energy shift is given by:

Defining the moment of inertia ^{8}

For O(

We can check (for

The mass gap does not lead to a new renormalization condition beyond Eq. (

In Ref. [

Further,
^{9}

Also

After introducing the renormalized couplings (

Note that the combination of the renormalized couplings is the same as one of the combinations appearing in Eq. (

Here we make a few remarks concerning the sensitivity of the observables on the four-derivative couplings

For a long cubic tube,

Note that all coefficients change sign as

Comparing

For the susceptibility calculated in

In the

Equation (

A specific feature of the

A derivation of the susceptibility from an

The absence of a

In Eq. (^{10}

The omitted terms at this order vanish exponentially as

Finally we remark that the case of two dimensions

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The

Note the identities

Completeness reads

From this we immediately get

For

Note that for

We start from the trivial identity

Let

So

For SU(3) we have using Eq. (

Now consider

For the SU(2) case the four-derivative operator

Consider the change of variables

By choosing

For the SU(2) case this corresponds to the change of variables

We have still to show that

Further, we can write

One has

Therefore we can conclude that the operator

A similar discussion to that presented above has been given by Leutwyler in Eq. (11.6) and the following paragraph of Ref. [

Consider the SU(

We parameterize

Define à la Faddeev–Popov

Now the action, measure, and also

The group volume is an irrelevant factor. Also, for DR we set

Now we only need

This induces a change

So

The zero-mode action is then given by

We have in particular

We give integrals with the Haar measure over SU(

It follows that

Note that for

The four-derivative terms

Using Eqs. (

Similarly for

We give some relations between different conventions for the O(4) couplings to connect with those used in Ref. [

Further,

From here

^{1} To avoid confusion with the box size

^{2} It is advantageous to treat these extra dimensions with a different size, since an extra check of the calculation is provided by the requirement that physical quantities are independent of this choice.

^{3} For SU(2) the identification to the O(4) fields

^{4} For

^{5} We used

^{6} The coefficients in Ref. [

^{7} Computation of higher coefficients

^{8} For

^{9} The expression in the square brackets above converges exponentially fast for

^{10} With the standard Hamiltonian proportional to the quadratic Casimir invariant