1 Introduction
Abstract

We study the renormalisation of gauge theories on general anisotropic lattices, to one-loop order in perturbation theory, employing the background field method. The results are then applied in the context of two different approaches to hadronic high-energy scattering. In the context of the Euclidean nonperturbative approach to soft high-energy scattering based on Wilson loops, we refine the nonperturbative justification of the analytic continuation relations of the relevant Wilson-loop correlators, required to obtain physical results. In the context of longitudinally rescaled actions, we study the consequences of one-loop corrections on the relation between the gauge theory and its effective description in terms of two-dimensional principal chiral models.

Renormalisation of gauge theories on general anisotropic lattices and high-energy scattering in QCD
Matteo Giordano***e-mail: giordano@atomki.mta.hu
Institute for Nuclear Research of the Hungarian Academy of Sciences
Bem tér 18/c, H-4026 Debrecen, Hungary
July 16, 2019

1 Introduction

Anisotropic lattices are a standard tool in modern lattice calculations, and have been used in the study of a large variety of problems, ranging from glueball [1] and light-hadron [2] spectroscopy to properties of QCD at finite temperature [3, 4]. Numerical calculations in four dimensions usually employ lattices with 3+1 anisotropy, i.e., only one of the lattice spacings is different from the others, while more general anisotropy classes have received much less attention [5], due to the increasing difficulty in the scale setting procedure. Indeed, for anisotropy classes other than 3+1, one needs to appropriately tune the action in order to recover Lorentz invariance in the continuum, already at the pure-gauge theory level. A better understanding of these more general anisotropy classes would be useful, since they provide a more flexible setting for varying length scales independently in different directions. This would allow, for example, to enlarge the range of momenta accessible to lattice calculation at a reasonable computational cost, by improving the resolution only in a single spatial direction [5].

Anisotropic lattices provide, quite obviously, the natural setting for the nonperturbative study of anisotropic systems, also beyond numerical applications. An interesting case is that of longitudinally rescaled actions, which in recent years have been considered in the context of high-energy scattering in QCD [6, 7, 8, 9, 10, 11, 12]. The basic idea of Refs. [6, 7, 8, 9] is to perform a rescaling of the longitudinal directions, which appear highly Lorentz-contracted in a high-energy scattering process, in order to derive an effective action starting from QCD. In Refs. [6, 7, 8, 9] only the classically rescaled action was considered, while the important effect of quantum corrections was studied later in Refs. [10, 11, 12], in the framework of Wilsonian anisotropic renormalisation in the continuum. In this context, the use of a gauge-invariant, anisotropic lattice regularisation could lead to more insight in the structure of quantum corrections. The relevant anisotropy class here is 2+2, with different lattice spacings in the longitudinal and in the transverse plane. This is also the case considered in Ref. [5], although for different purposes.

In Ref. [13] a classical anisotropic rescaling of the functional integral has been used to justify, on nonperturbative grounds, the analytic continuation from Euclidean to Minkowski space, required to obtain physical results in the Euclidean formulation [14, 15, 16, 17, 18, 19] of the nonperturbative approach to soft high-energy scattering [20, 21, 22, 23, 24, 25, 26, 27]. This approach has been recently used in Ref. [28] to obtain a theoretical estimate of the leading energy dependence of hadronic total cross sections, resulting in fair agreement with experiments. As the analytic continuation plays a key role in this approach, it is important to establish its correctness going beyond the formal argument of Ref. [13], which, as we have said above, is based only on a classical rescaling of the QCD action. To this end, quantum corrections to the effective action must be included to prove that the necessary analyticity requirements are actually fulfilled. The relevant anisotropy class in this case is 2+1+1, with different lattice spacings in the transverse plane and in the two longitudinal directions.

The purpose of this paper is to perform the renormalisation of a gauge theory regularised on a general anisotropic lattice, and to apply the results in the study of hadronic high-energy scattering through the approaches mentioned above. To avoid the complications related to the introduction of fermions on the lattice, we work here in the quenched approximation, i.e., pure gauge theory.

The plan of the paper is the following. In Section 2 we study renormalisation for a general anisotropic lattice regularisation, using the background field method on the lattice [29, 30, 31, 32, 33, 34, 35, 36, 37]. In Section 3 we use the results in the context of the nonperturbative approach to soft high-energy scattering of Refs. [20, 21, 22, 23, 24, 25, 26, 27], refining the argument of Ref. [13] on the possibility of performing analytic continuation to Euclidean space. In Section 4 we discuss the longitudinally rescaled actions of Refs. [6, 7, 8, 9, 10, 11, 12], focussing on the representation of the gauge theory as a set of coupled two-dimensional principal chiral models. Finally, Section 5 contains our conclusions and prospects. Some technical details are discussed in the Appendices.

2 Anisotropic renormalisation

Our aim is to renormalise the Euclidean gauge theory regularised on a 4D orthogonal anisotropic lattice. More precisely, lattice points are located at , where are four orthogonal unit vectors, and the physical coordinates in Euclidean space are , . Here is the lattice spacing in direction , with the dimensionless anisotropy parameters being the inverse ratios of to a reference length scale .111The five parameters and are obviously redundant, and some condition has to be imposed on to remove this redundancy. This notation is however convenient, as we treat all the directions on the same footing. Consider the following Wilson-like action,

 Streelat=β∑n∑μ<νCμν(1−1NcRetrUμν(n))=β∑n∑μ<νCμνPμν(n), (2.1)

where are the usual plaquette variables built up with the link variables ,

 Uμν(n)=Uμ(n)Uν(n+^μ)U†μ(n+^ν)U†ν(n), (2.2)

with the coupling constant, and , , are the plaquette coefficients,222For definiteness, we define also .

 Cμν=Cνμ=Cμν(λ)=(λμλν)2J,J−1≡4∏α=1λα. (2.3)

It is straightforward to show that Eq. (2.1) yields the correct naïve continuum limit upon identification of the continuum, physical gauge fields through

 Uμ(n)=eigaμAμ(x(n)), (2.4)

as appropriate for an anisotropic lattice. The choice of plaquette coefficients is easily understood by noticing that is just the volume of an elementary cell, so that is the Jacobian for the change of variables from isotropic to anisotropic coordinates, while is the area of the faces of an elementary cell lying in the plane.

As is well known, divergencies appear in the continuum limit when taking into account quantum corrections. These divergencies need to be subtracted through a suitable renormalisation of the couplings in order to obtain a finite continuum theory. On the isotropic lattice, the symmetry under the unbroken hypercubic subgroup of guarantees that all the plaquette terms in the action need to be renormalised in the same way, so that a single redefinition of is sufficient to reabsorb the divergencies. The form of the action is therefore unchanged, and one recovers the full invariance in the continuum limit.

On a general anisotropic lattice this residual symmetry is broken, except for reflections through lattice hyperplanes, and so in general different terms will require a different renormalisation. Since there are six different plaquette terms and only four lattice spacings, it will not be possible in the general case to reabsorb completely the quantum corrections into a redefinition of , keeping at the same time the same form of the tree-level action [5]. In turn, this implies that the continuum limit of Eq. (2.1) cannot be made into an -invariant theory by an appropriate, simple rescaling of the lattice spacings, since in the general case one will still find different coefficients for the six continuum field-strength terms. To recover invariance one must ensure that these coefficients are equal, and this requires that we take the action to be of the more general form

 Slat=∑n∑μ<νβμνCμν(1−1NcRetrUμν(n))=∑n∑μ<νβμνCμνPμν(n), (2.5)

where the couplings have to be properly tuned to yield a finite, -invariant theory in the continuum limit.

The need for tuning comes, as we have said, from the fact that there are in general more couplings than anisotropy parameters. It is however easy to show that one has to tune at most only two combinations of the couplings to achieve restoration of invariance in the continuum limit, while the other four independent combinations can be interpreted as the coupling fixing the overall lattice scale, and renormalisations of the anisotropies . To see this, let us remove the redundancy in the set by imposing the symmetric condition , thus defining in terms of the volume of an elementary cell. Any other equivalent choice (i.e., giving the same ) is obtained by a simple global rescaling of and of . The six plaquette terms can be grouped in pairs of “complementary” and plaquettes, i.e., , etc., which we denote as . It is also easily noticed that , so that . This suggests to parameterise as follows,

 βμν=βZ(μν|¯μ¯ν)zμzνz¯μz¯ν. (2.6)

As there are two redundant parameters, we choose to fix , so that our condition on is not renormalised,333Any other choice is of course allowed. If, for example, the scale is defined to be one of the lattice spacings by choosing for some , then it is convenient to choose . The new values of are obtained from those corresponding to the symmetric condition by replacing , while and are unaffected. and . In this way , and are unambiguously defined and can be obtained from as follows,

 β=(∏μ<νβμν)16,zμ=⎛⎝∏ν≠μβμνβ¯μ¯ν⎞⎠18,Z(μν|¯μ¯ν)=⎡⎢⎣βμνβ¯μ¯ν(βμ¯νβ¯μνβμ¯μβν¯ν)12⎤⎥⎦13. (2.7)

This makes it explicit that the restoration or not of invariance in the continuum depends only on the values of the ratios of the couplings . Defining now the bare anisotropy parameters , and the bare plaquette coefficients , one can rewrite Eq. (2.5) as

 Slat=β∑n,(μν|¯μ¯ν)Z(μν|¯μ¯ν)[CBμνPμν(n)+CB¯μ¯νP¯μ¯ν(n)]. (2.8)

This equation shows that to obtain an -invariant theory in the continuum limit, one can choose freely (up to a constraint to remove the redundancy), and then tune only the two independent ratios of to the appropriate values. The physical anisotropy parameters are related to the bare ones through the renormalisation , and can be measured ex post.

Using the parameterisation Eq. (2.7), it is possible to set up a rather simple nonperturbative scheme to achieve restoration of invariance in the continuum, for an arbitrary choice of bare anisotropy parameters. The basic idea is to impose that the string tension, determined from the asymptotic behaviour of large rectangular on-axis Wilson loops , is the same for all pairs of directions . Denoting with the physical (dimensionful) string tension, this amounts to impose , for all pairs of different . Multiplying the relations for and its “complementary” , one obtains the following consistency conditions,

 ^σ12^σ34=^σ13^σ24=^σ14^σ23, (2.9)

which have to be imposed to recover invariance. This can be done without any prior knowledge of the physical , and requires only to properly tune two of the coefficients (the third one being constrained by our choice ). Having done this, the anisotropies can then be obtained from the ratio for any . Imposing one can explicitly determine all ’s, and set the lattice scale from the relation . While the string tension is known not to be the best observable for setting the physical scale, nevertheless it could be useful for the tuning, as it can be determined to high precision by means of multilevel algorithms [38]. It is worth mentioning that the tuning of two parameters is only required when all the lattice spacings are different: if at least a pair of lattice spacings are equal, one easily sees that only one parameter has to be tuned.444 In the 3+1 case, where a single lattice spacing differs from the others, there are only two kinds of plaquette terms and so only two independent lattice string tensions. In this case there is thus no consistency condition to be satisfied and no tuning is needed, as is well known.

2.1 Background field method

From the discussion above, we see that our task is to find the relations among the couplings that will lead to an -invariant theory in the continuum limit. We will study this problem to lowest order in perturbation theory, making use of the background field method [29, 30, 31, 32] on the lattice [33, 34, 35, 36, 37]. The advantage of this method is that it allows to keep an exact gauge invariance on the lattice after gauge fixing, which greatly simplifies the calculations. A full account on the background field method can be found elsewhere [39, 40]. Here we briefly recall the main points of the method to fix the notation.

The first step is to introduce a background field in the gauge action as follows,

 SBF[U(c),V]≡Slat[VU(c)], (2.10)

where we now denote with the gauge links, to be integrated over with the usual Haar measure. As a consequence of the gauge invariance of , the action is invariant under the background gauge transformation

 U(c)Gμ(n)=G(n)U(c)μ(n)G†(n+^μ),VGμ(n)=G(n)Vμ(n)G†(n), (2.11)

with , as well as under the following gauge transformation of alone,

 Vμ(n)→G(n)Vμ(n)U(c)μ(n)G†(n+^μ)U(c)μ†(n). (2.12)

The integration measure is also obviously invariant under the transformations Eqs. (2.11) and (2.12). One then proceeds to set up perturbation theory in the usual way, setting

 Vμ(n)=eigλμqμ(n),U(c)μ(n)=eiaμBμ(n), (2.13)

where555Here and in the following, the sum over repeated colour indices is understood. and , with the generators of in the fundamental representation, , and . One then changes variables of integration to , expressing the Jacobian as a contribution to the action. Notice that powers of and are chosen so that is dimensionless, while has dimensions of mass. This distinction is convenient for book-keeping purposes [36].

Under the transformation Eq. (2.11), the background field transforms as a gauge field, while the “quantum” field transforms as a matter field in the adjoint representation. The symmetry under the gauge transformation Eq. (2.12) requires to impose a gauge condition on to define the corresponding propagator. This is done à la Faddeev–Popov, adding a gauge-fixing term to the action, together with the corresponding ghost term. The key point is that there is an appropriate choice of gauge, called the background field gauge, for which the gauge-fixing and the ghost terms are invariant under the background gauge transformation Eq. (2.11). This gauge-fixing term is [29, 30, 37]

 Sg.f.[B,q]=J∑ntr(∑μD−μqμ)2, (2.14)

where are the lattice background covariant differences,

 D+μf(n) ≡λμ[U(c)μ(n)f(n+^μ)U(c)μ†(n)−f(n)], (2.15) D−μf(n) ≡λμ[U(c)μ†(n−^μ)f(n−^μ)U(c)μ(n−^μ)−f(n)],

in which a factor is also included for convenience. The usual lattice differences are obtained setting in the expressions above, where denotes the unit matrix. The corresponding ghost term is

where , , with independent Grassmann variables, and where

It is straightforward to prove invariance of these two terms under the background gauge transformation, Eq. (2.11), supplemented by the transformation laws for the ghost fields,

 cG(n)=G(n)c(n)G†(n),¯cG(n)=G(n)¯c(n)G†(n). (2.18)

Expanding Eq. (2.16) up to , one finds

 Sghost[B,q,c,¯c] =2J∑n,μtr{[D+μ¯c(n)][D+μc(n)]}+O(g) (2.19) =2J∑n,μtr{¯c(n)D−μD+μc(n)}+O(g)≡S0ghost[B,c,¯c]+O(g),

where we have used “integration by parts” on a lattice (infinite or with periodic boundary conditions),

 (2.20)

The starting point for the perturbative analysis is the generating functional

 Z[B,J,¯η,η] =∫DqDcD¯ce−Stot[B,q,c,¯c]+J⋅q+¯η⋅c+η⋅¯c=eW[B,J,¯η,η], (2.21) Stot[B,q,c,¯c] =SBF[B,q]+Smeas[q]+Sg.f.[B,q]+Sghost[B,q,c,¯c],

where with a small abuse of notation we have written , and we have added source terms for the various fields. Here and , and similarly for the other terms. A Legendre transform gives the effective action (generating functional for 1PI graphs),

 Γ[B,Q,C,¯C]=−W[B,J,¯η,η]+J⋅Q+¯η⋅C+¯C⋅η, (2.22)

where the classical fields , and are defined as

 Qaμ(n)=∂W[B,J,¯η,η]∂Jaμ(n),Ca(n)=∂W[B,J,¯η,η]∂¯ηa(n),¯Ca(n)=∂W[B,J,¯η,η]∂ηa(n), (2.23)

i.e., they are the expectation values of the quantum fields for prescribed values of and of the sources.

Defining a background gauge transformation for the classical fields, imposing that they transform as the corresponding quantum fields, Eqs. (2.11) and (2.18), leads finally to the identity

 Γ[BG,QG,CG,¯CG]=Γ[B,Q,C,¯C] (2.24)

for the effective action. This is the key relation that allows us to simplify the calculations. Indeed, setting , as a consequence of the background gauge invariance, of the discrete symmetries of the action (translations and reflections666Reflections act as follows on the coordinates, for , . The corresponding transformation laws for and are the following, (2.25) ), and of the locality of divergencies, to one-loop accuracy and to lowest order in perturbation theory we are guaranteed to find in the continuum limit

 lima→0Seff[B]= 12∑μ,ν∫d4x[βμν2Nc−Kμν]trF2μν(x) (2.26) auauauau+(non-local finite terms)+O(g2),

where is , and where is the field strength for the continuum background field , . For our purposes it is therefore sufficient to compute the two-point function of the background field to have enough information to renormalise the theory and impose invariance. To one-loop accuracy it is enough to set

 βμν2Nc−Kμν=1g2+δβμν2Nc−Kμν=1g2r, (2.27)

where is the renormalised, -independent coupling.

We notice that Eq. (2.5), with the couplings chosen according to Eq. (2.27), can be interpreted in two ways. Under the identification with , it leads in the continuum to the renormalised, isotropic action for the gauge fields , for which it provides an appropriate lattice discretisation. On the other hand, identifying with , in the continuum limit one obtains the following renormalised anisotropic action,

 S→12g2r∑μ,νCμν∫d4ytrΦ2μν(y), (2.28)

with the usual field-strength tensor for , for which Eq. (2.5) provides therefore a lattice discretisation. This is the form of the action obtained by classically rescaling coordinates and fields in the Yang-Mills action, discussed in Refs. [6, 7, 8, 9, 13].

2.2 One-loop calculation

To compute it is enough to expand the action to order , which in turn means expanding the gauge action up to second order in . Contributions from are at least and can be ignored. Let us expand the action in powers of ,

 SBF[B,q]+Sg.f.[B,q]=Sc[B]+Sg1[B,q]+Sg2[B,q]+…, (2.29)

where is the classical action, is linear in , is quadratic and so on, and set

 S2[B,q,c,¯c] =Sg2[B,q]+S0ghost[B,c,¯c] (2.30)

A straightforward calculation then shows that

 Seff[B]∣∣O(g0) =SBF[B,0]+12logdetΠ[B]detΠ[0]−logdet^Π[B]det^Π[0]. (2.31)

Terms linear in play no role and can be ignored.777These terms are usually discarded by requiring to satisfy the equations of motion, but this is actually not necessary. Eq. (2.31) can be conveniently written as

 e−Seff[B]∣∣O(g0)=e−Sc[B]⟨e−(S2−Sfree)⟩0, (2.32)

where is the free action with no background field, and denotes the corresponding expectation value,

 ⟨O[B,q,c,¯c]⟩0 =Z−1free∫DqDcD¯ce−Sfree[q,c,¯c]O[B,q,c,¯c], (2.33) Zfree =∫DqDcD¯ce−Sfree[q,c,¯c].

For future utility, we define the connected correlation function . Since we are interested only in the two-point function for , only terms up to will be kept in .

2.2.1 The quadratic action

The gauge action in a background field can be conveniently written as follows,

 SBF[B,q]=Slat[VU(c)]=∑n,μ<νβμνCμν(1−1NcRetr{Vμν(n)U(c)μν(n)}), (2.34)

where the “quantum” and the “background” plaquette are given respectively by

 Vμν(n) ≡e−ig1λμ(1λνD+νqμ(n)+qμ(n))e−ig1λνqν(n)eig1λμqμ(n)eig1λν(1λμD+μqν(n)+qν(n)), (2.35) U(c)μν(n) ≡U(c)μ(n)U(c)ν(n+^μ)U(c)μ†(n+^ν)U(c)ν†(n).

A standard application of the Baker-Campbell-Hausdorff formula gives

 U(c)μν(n)=exp{ia2λμλνfμν(n)+O(a3B∂B,a4(∂B)2)+O(a3B3)}, (2.36)

with888In the following equations we will sometimes drop the dependence on the lattice site to make the expressions more readable.

 fμν=a−1(Δ+μBν−Δ+νBμ)+i[Bμ,Bν], (2.37)

which in the continuum limit reduces to the usual field strength tensor for the background field.999 In principle, also the higher-order terms of order appearing in Eq. (2.36) could contribute to the two-point function in the continuum. This however is not the case, as we will see below (see footnotes 10 and 11). For we have instead

 Vμν(n)=exp{ig1λμλν[Fμν(n)+gRμν(n)]+O(g3)}, (2.38)

where

 Fμν =D+μqν−D+νqμ, (2.39) R(1)μν =i2λμλν[D+μqν,D+νqμ]+i[qμ,qν], R(2)μν =i2(1λμ[qμ,D+νqμ]−1λν[qν,D+μqν]),

and . Expanding up to quadratic terms in and we find

 SBF[B,q]=Sc[B]+Sq[B,q]+(linear in q)+O(q3), (2.40)

where is the classical action, already defined above,

 Sc=Ja4∑n,μ,νβμν2Nc12trf2μν(n)→a→012∫d4x∑μ,νβμν2NctrF2μν(x), (2.41)

while the “quantum” piece is given by

 Sq[B,q]=J∑n,μ,ν12tr{F2μν(n)+2a2Rμν(n)fμν(n)}−14a4(λμλν)2tr{F2μν(n)f2μν(n)}. (2.42)

The gauge-fixing term is quadratic in , and can be conveniently rearranged as follows,

 Sg.f.=S(1)g.f.+S(2)g.f.+ST′, (2.43)

where

 S(1)g.f. =J∑n,μ,νtr{D+νqμ(n)D+μqν(n)},S(2)g.f.=Ja2∑n,μ,νtr{¯R(1)μν(n)fμν(n)}, (2.44) ¯R(1)μν =i([qμ,qν]+1λμ[qμ,D+μqν]−1λν[qν,D+νqμ]+1λμλν[D+νqμ,D+μqν]), ST′ =Ja4∑n,μ,νtr¯R(2)μν(n) ¯R(2)μν =i2λμλν[fμν,1λμD+μqν+qν][fμν,1λνD+νqμ+qμ].

Finally, the ghost term is independent of to . Putting all the terms together, one obtains for the quadratic lattice action

 S2=Sc+Sfree+Sintgluon+Sintghost+SA+SB+ST+ST′, (2.45)

where the terms have been grouped so that each quantity in the equation above is separately invariant under a background gauge transformation [36]. Here , with

 (2.46)

being the free actions for gluons and ghosts, respectively, in terms of which the propagators are defined, while the interaction terms are given by

 Sintgluon =Sq+S(1)g.f.−Sfreegluon=J∑n,μ,νtr{[D+μqν(n)]2−[Δ+μqν(n)]2}, (2.47) Sintghost =S0ghost−Sfreeghost=2J∑n,μtr{¯c(n)[D−μD+μ−Δ−μΔ+μ]c(n)}.

Moreover, extra vertices come from the terms

 SA (2.48) SB =Ja2∑n,μ,νtr{R(2)μν(n)fμν(n)},ST=−Ja4∑n,μ,ν1(2λμλν)2tr{F2μν(n)f2μν(n)}.

Explicitly, we have for and the expressions101010 In these quantities one should in principle include also the higher-order terms mentioned above in footnote 9, by properly redefining .

 SA =Ja2∑n,μ,ν12tr{Aμν(n)fμν(n)}, (2.49) Aμν SB =Ja2∑n,μ,ν12tr{Bμν(n)fμν(n)}, Bμν =i(1λμ[qμ,D+νqμ]−1λν[qν,D+μqν]).

Notice that the terms and are odd in a given component of the gluon field, while the other terms are even. Since the propagator is diagonal, this implies [33, 35, 36] that111111 This clearly remains true also if higher-order terms neglected in Eq. (2.36) are included in the definition of , see footnotes 9 and 10. Since is , the only contribution of a higher-order term which should still be considered is that of the term in Eq. (2.36) to ; however, (global) background gauge invariance implies that , and so higher-order terms can be safely ignored. , and also that