Renormalization group, trace anomaly and Feynman-Hellmann theorem

We show that the logarithmic derivative of the gauge coupling on the hadronic mass and the cosmological constant term of a gauge theory are related to the gluon condensate of the hadron and the vacuum respectively. These relations are akin to Feynman-Hellmann relations whose derivation for the case at hand are complicated by the construction of the gauge theory Hamiltonian. We bypass this problem by using a renormalisation group equation for composite operators and the trace anomaly. The relations serve as possible definitions of the gluon condensates themselves which are plagued in direct approaches by power divergences. In turn these results might help to determine the contribution of the QCD phase transition to the cosmological constant and test speculative ideas.


INTRODUCTION
The Feynman-Hellmann theorem relates the leading order variation of the energy to a local matrix element, providing a direct link between an observable and a theoretical quantity. Originally derived in quantum mechanics, its application to quantum field theory (QFT) is generally straightforward and widely used, see e.g. [1]. Exceptions are cases where the Hamiltonian is difficult to construct, which may arise as a QFT is usually defined from a Lagrangian where (most) symmetries are manifest. An example of such a case are gauge theories where the elimination of two degrees of freedom from the four vector A µ are at the root of the problem. In this work we bypass the construction of the gauge field part of the Hamiltonian by using renormalisation group equations (RGE) for composite operators as well as the trace anomaly. We obtain a relation that relates the logarithmic derivative of the hadron mass with respect to the coupling constant, and the gluon condensate of the hadron state. Likewise we find a similar relation relating the derivative of the cosmological constant and the vacuum gluon condensate.
The corresponding relation for the quark mass were used in Ref. [2] to derive the leading scaling behaviour of the hadronic masses for a non-trivial infrared (IR) fixed point, deformed by the fermion mass parameter. The relation derived in this paper are used to compute the scaling corrections to the hadronic mass spectrum [3].

RGE for local composite operators
We begin by defining the relation between the bare operator O and the renormalized operatorŌ: where (g, m) are the bare gauge coupling and mass, Λ is the UV cut-off of the theory, (ḡ,m) are the set of renormalized couplings and µ is the renormalisation scale. From one obtains an RGE of the form, Denoting by d O ≡ dŌ the engineering dimension of O one gets by dimensional analysis can be solved by the method of characteristics by introducing a parameter which has the interpretation of a blocking variable. This is for instance used in Ref. [3] to identify the scaling corrections to correlators at a nontrivial IR fixed point.

Trace anomaly
Let us first define some conventions. The Lorentzinvariant normalisation of states for D-dimensional space-time is given by the diagonal matrix elements are abbreviated to where c denotes the connected part and H stands for any physical state. Since the energy momentum tensor is related to the four momentum operatorp µ = d D−1 xT 0µ it is readily seen that: where O 0 ≡ 0|O|0 hereafter with |0 denoting the vacuum state, p µ is the four momentum associated with the physical state (E = p 0 ), g µν is the Minkowski metric with signature (+, −, ..., −) and Λ GT is the cosmological constant contribution of the gauge theory under consideration.
The traces of the right hand side (RHS) are the masses of the hadrons 2M 2 H and the masses density of empty space DΛ GT = g µ µ Λ GT , and the left hand sides (LHS) follow from the trace anomaly. For a gauge theory with N f Dirac quarks the trace anomaly, in terms of renormalized fields, is given by [4] [13], where the subscript "on-shell" (10) indicates that the equation, in this form, is to be used on the physical subspace only. Furthermore we have introduced the following shorthand notation: with G A αβ being the gauge field strength tensor and A a colour index. Summation over indices is understood. Conventions are such that the coupling is absorbed into the gauge field. We note that the trace energy momentum tensor is not renormalized ( i.e. T µν =T µν ) as it is directly related to the four momentum which is a physical quantity. FurthermoreQ = Q is an RG invariant which then implies thatβ/(2ḡ)Ḡ +γ mQ is an RG invariant on the subspace of physical states. In particular we noteḠ = G which is of importance when interpreting our final result. Finally Eqs. (10) and (9) lead to:

Feynman-Hellmann theorem in QFT
Let us start by recalling the main steps in the derivation of the Feynman-Hellmann theorem in quantum mechanics, which is a simple but powerful relation which has been obtained by a number of authors [5]. Let us consider a quantum-mechanical system, whose dynamics is determined by a Hamiltonian H(λ), which depends on some parameter λ. The Feynman-Hellmannn theorem states that the λ-derivative of the energy equals the derivative of the Hamiltonian when evaluated on the corresponding eigenstates: It relies on the observation that The adaption to QFT, in the simplest cases, necessitates solely to take into account the relativistic state normalisation (7). E.g. with H m = Q =Q (11), where H tot = H m + ..., the Feynman-Hellmann theorem for the mass reads:m In (16) (14) on the LHS originates from the additional factor of 2E H in the normalisation (7). In (17) the normalisation 0|0 = 1 was assumed. Furthermore we note that in a mass independent scheme ( ∂ ∂mZ m = 0) m ∂ ∂m = m ∂ ∂m and since Q =Q, therefore the relation (16) also holds for bare quantities. Eq. (16) is widely known and used in lattice simulation to extract the corresponding contribution to the nucleon mass for example. Noting that m ∂ ∂m E 2 H = m ∂ ∂m M 2 H [14], it follows that Q EH = Q MH with normalisation (7) is a static quantity.

GLUON CONDENSATES THROUGH RGE AND FEYNMAN-HELLMANN THEOREM
The adaption of the analoguous relations (16,17) with regard to the gauge coupling g is complicated by the fact that in gauge theories the construction of the Hamiltonian itself is rather involved, see e.g. Ref. [6]. As stated in the introduction, we bypass the construction of the Hamiltonian, and its primary and secondary constraints, by using the RGE, the trace anomaly, and the relations for the mass.
The RGE (6) for the M 2 H and Λ GT [15][16] (20) and from there we read off our main results, For the first relation we have used ∂ ∂ḡ M 2 H = ∂ ∂ḡ E 2 H , where the same remark applies as for the derivative with respect to m given earlier on. In particular this implies that Ḡ EH = Ḡ MH is a static quantity[17]. The scheme dependence of the gluon condensates, inherent in the earlier statementḠ = G, is made manifest throughḡ ∂ ∂ḡ = g ∂ ∂g and the fact that E 2 H and Λ GT , being physical quantities, do not renormalize. Thus we wish to stress that it is vital to distinguish bare and renormalized quantities when discussing the relations (21,22). The condensates may be computed with lattice Monte Carlo simulation in some fixed scheme. The conversion to other schemes, say, the MS-scheme can be done through a perturbative computation at some large matching scale. For example defining two schemes a and b through g = Z aḡa = Z bḡb one gets: (23) The transformation between scheme a and b is therefore given by Ḡ a = Z ab G Ḡ b according to Eqs. (21,22) for both the vacuum and particle gluon matrix element. The derivation of the relations (21,22) with bare couplings would surely be possible, but we do not consider it a necessity.
It is worthwhile to illustrate the importance of using eigenstates of the Hamiltonian for the matrix elements considered in the Feynman-Hellmann theorem by an example at hand. One might be tempted to obtain the relation (16) directly from the trace anomaly (10) assuming a mass independent scheme (which entails thatβ andγ m are independent of m) viā which without corrections andγ m = 1 contradicts (16). The necessary corrections originate from the fact that T µ µ does not commute with the Hamiltonian in general and therefore is not an eigenoperator of the physical states (7). Thus differentiation of the states with respect tom ∂ ∂m is required for consistency and exemplifies the importance of the energy eigenstates in the Feynman-Hellmann theorem.

CONCLUSIONS AND DISCUSSION
In this work we have derived relations between the logarithmic derivative of the mass of a state (and the vacuum energy) with respect to the gauge coupling in terms of the corresponding gluon condensates. These relations are summarized in Eqs. (21,22). We shall comment on the interest of these equations for various aspects in the paragraphs below.
First the lnḡ-derivative of M 2 H and Λ GT may be taken as a definition of the gluon condensates. This means that the LHS, computable in lattice Monte Carlo simulations, serves as a definition of the condensates on the RHS. For Ḡ MH this way of computing seems to entail the advantage that the disconnected part, related to difficulties in direct computations, is automatically absent as they do not contribute to M 2 H on the LHS. The scheme dependence of the condensate is determined by the scheme dependence of the LHS and has been discussed in the text. The transition from one scheme to another can be achieved by a perturbative computation provided the matching scale is high enough for perturbation theory to be valid.
We shall add a few remarks on the gluon condensates. In QCD the matter condensatesβ/(2ḡ) Ḡ EH are known indirectly through the mass (Eq. (12)) for light mesons, other than the pseudo Goldstone bosons π, K, η.., as for the latterQ is negligible since it is O(m light ). For the nucleon this was first discussed in [8]. For the B-meson β/(2ḡ) Ḡ EH is related to a non-perturbative definition of the heavy quark scale Λ HQ [9]. The determination of the gluon vacuum condensate is of importance for QCD sum rules [10] as well as for the cosmological constant problem to be discussed further below. The value of the gluon condensate cannot be regarded as settled. This is, in part, due to the fact that there is no direct first principle determination of the gluon condensate.
Let us comment on aspects of the cosmological constant, which is a topic of more speculative nature. Without gravity only energy differences matter. Thus the cosmological constant is only determined up to a constant in flat space. Yet the difference of the cosmological constant due to the QCD phase transition itself is generally seen to be a tractable quantity, given by Eq. (13) provided the condensates are well defined. The quark condensate is known through the Gell-Mann Oakes Renner relation: with m π and f π being the pion mass and decay constants. It would seem that any undetermined constant of the gluon condensate should drop out in Eq. (22). Therefore Ḡ 0 determined from this equation could be reinserted into Eq. (13), where scheme dependence cancels provided the appropriateβ andγ m are used [18]. Scheme independence in turn might be used as a consistency check of the ideas brought forward in this paragraph.
At last let us add that if the gluon condensate can be determined, then it could be checked to what degree the lowest J PC = 0 ++ -state in a confining gauge theory saturates the partial dilation conserved current hypothesis, see e.g. Ref. [11]. This could serve as a quantitative measure to identify what is commonly referred to as a dilaton in the literature. The possibility that the Higgs boson candidate discovered at the LHC might be a dilaton of a gauge theory with slow running coupling (walking technicolor) is a possibility that is still considered within the particle physics community e.g. [12].
Acknowledgements: RZ acknowledges the support of advanced STFC fellowship. LDD and RZ are supported by an STFC Consolidated Grant. m denotes the fixed point value of γm). This observation was essentially already made in our previous paper [7] by deriving Q 0 ∼ G 0 ∼ m D/(1+γ * m ) .
[16] For pure Yang Mills (YM) theory, where effectively Q → 0, we note that (18) leads to (βYM) −1 = − ∂ ∂ḡ ln MH with H being a glueball state. This may serve as a definition ofβYM. The extension of this idea to a gauge theory with fermions is not immediate.
[17] We could have obtained this result earlier on by inserting (12) into (16) and using Q M H = Q E H .
[18] A more complete discussion would include the mass degeneracy of the flavours. The effect of heavy flavours (m quark > ΛQCD) can in principle be absorbed into the beta function. The precise discussion of which goes beyond the scope of this letter.