b

anifestly

angiansof localmetry

baryonsfieldtion of[14] as

rmulatedvariant

odifiedryad to ants. Themmation

Physics Letters B 575 (2003) 11–17

www.elsevier.com/locate/physlet

Power counting in baryon chiral perturbation theoryincluding vector mesons

Thomas Fuchsa, Matthias R. Schindlera, Jambul Gegeliaa,b,1, Stefan Scherera

a Institut für Kernphysik, Johannes Gutenberg-Universität, D-55099 Mainz, Germanyb High Energy Physics Institute, Tbilisi State University, University St. 9, 380086 Tbilisi, Georgia

Received 6 August 2003; accepted 19 September 2003

Editor: J.-P. Blaizot

Abstract

It is demonstrated that using a suitable renormalization condition one obtains a consistent power counting in mLorentz-invariant baryon chiral perturbation theory including vector mesons as explicit degrees of freedom. 2003 Published by Elsevier B.V.

PACS:11.10.Gh; 12.39.Fe

1. Introduction

Because of their phenomenological importance, vector mesons were included in low-energy chiral Lagralready at an early stage [1–3]. Usually they were treated—within some approximation—as gauge bosonschiral symmetry. Refs. [4,5] contain reviews of these approaches including one of a “hidden” local chiral sym(see also Ref. [6]). Details of the Lagrangian describing the interaction of vector mesons with mesons andin a chirally invariant way can be found in Refs. [7–9]. Different formulations of vector meson effectivetheories were shown to be equivalent in Ref. [10]. However, to the best of our knowledge the incorpora(axial) vector mesons into (baryon) chiral perturbation theory [11–13] remains an important open problemlong as a systematic power counting is not available. In the present Letter we show how the approach forecently in Ref. [15] is capable to consistently include (axial) vector mesons in the manifestly Lorentz-informulation of the effective field theory of the strong interactions.

The basic idea of Ref. [15] (see also Refs. [16,17]) can be summarized as follows. If one uses the mminimal subtraction scheme of (baryon) chiral perturbation theory (MS) [13], then the diagrams with an arbitranumber of loops contribute to lower-order calculations. As mentioned in Ref. [13], these contributions lerenormalization of the low-energy constants, i.e., they can be absorbed into a redefinition of these constarenormalized coupling constants of our extended on-mass-shell (EOMS) scheme correspond to a resu

E-mail address:gegelia@kph.uni-mainz.de (J. Gegelia).1 Alexander von Humboldt Research Fellow.

0370-2693/$ – see front matter 2003 Published by Elsevier B.V.doi:10.1016/j.physletb.2003.09.060

12 T. Fuchs et al. / Physics Letters B 575 (2003) 11–17

inlingsulations.nclusionr only as

We will

ven by

y

d chiral

on. The

r

son loop

2)

of the (infinite) series of loop corrections. The coupling constants and fields of theMS scheme are expressedterms of EOMS quantities. ExpandingMS quantities in terms of a power series of EOMS-renormalized coupgenerates counterterms which precisely cancel the contributions of multi-loop diagrams to lower-order calcThe EOMS scheme thus leads to a consistent power counting in baryon chiral perturbation theory. The iof (axial) vector mesons into this scheme does not introduce any new complications as long as they appeainternal lines in Feynman diagrams involving soft external pions and nucleons with small three momenta.show this below by means of three select examples.

2. Effective Lagrangian and power counting

In our discussion we will make use of the effective Lagrangian (including vector mesons) in the form giWeinberg [3],

L= 1

2∂µπ

a∂µπa − M2

2πaπa + Ψ

(iγ µ∂µ − m

)Ψ − 1

4FaµνF

aµν + 1

2M2

ρAaµA

aµ

(1)+ gεabcπa∂µπbAcµ + gΨ γ µ τa

2ΨAa

µ − gA

2FΨ γ µγ5τ

aΨ ∂µπa +L1,

whereπa andAaµ are isotriplets of pion andρ meson fields with massesM andMρ , respectively, andΨ is

an isodoublet of nucleon fields with massm. The constantsF and gA denote the chiral limit of the pion decaconstant and the axial–vector coupling constant, respectively. Moreover, we use the universalρ coupling, i.e.,g = gρππ = gρNN . The field strengths are defined asFa

µν = ∂µAaν − ∂νA

aµ + gεabcAb

µAcν andL1 contains an

infinite number of terms.Our renormalization scheme is devised so that, after renormalization, a given diagram has a so-calle

powerD [18] which is determined by applying the following power counting rules. LetQ collectively stand forthe pion mass, a (small) external four-momentum of a pion or a (small) external three-momentum of a nuclepion nucleon interaction counts asO(Q), theρ nucleon interaction asO(Q0), theππρ interaction asO(Q), andtheρ self-interaction terms asO(Q0), respectively. The integration of a loop counts asO(Qn) (in n dimensions).Finally, we count theρ meson propagator asO(Q0), the nucleon propagator asO(Q−1),2 and the pion propagatoasO(Q−2), respectively.

3. Applications

Let us now consider three characteristic diagrams illustrating our approach. We start with the vector-mecontribution to the nucleon self energy (see Fig. 1):

(2)−iΣ1(/p) = −g2

4

∫dnk

(2π)nγντ

a 1

/p − /k − m + iεγµτ

agµν − kµkν

M2ρ

k2 − M2ρ + iε

.

According to the above power countingΣ1 is assigned the chiral orderQn−1. Calculating the expression of Eq. (we obtain:

(3)Σ1 = − 3g2

8M2ρp

2

[AIρ(0) + BIN(0) + CIρN(0,−p)

],

2 Fermion loops are integrated out, their contributions being included in low-energy constants.

T. Fuchs et al. / Physics Letters B 575 (2003) 11–17 13

rly

tionor this

rl toat

terms of

ponding

5)

Fig. 1. One-loop contribution to the nucleon self energy due toρ meson dressing.

where

A = /p[p2 − m2 + (2− n)M2

ρ

],

B = /p[p2 + m2 − (2− n)M2

ρ

] − 2p2m,

C = /p[(3− n)M2

ρ

(p2 + m2) − (2− n)M4

ρ − (p2 − m2)2

]+ 2(n − 1)M2

ρp2m,

and

Iρ(0) = i

∫dnk

(2π)n

1

k2 − M2ρ + iε

,

IN (0) = i

∫dnk

(2π)n

1

k2 − m2 + iε,

(4)IρN(0,−p) = i

∫dnk

(2π)n

1

[k2 − M2ρ + iε][(k − p)2 − m2 + iε] .

Before renormalization, the expression of Eq. (3) forΣ1 is of orderQ0, i.e., the unrenormalized diagram cleaviolates the power counting rules.

To renormalizeΣ1 we first apply theMS scheme (modified minimal subtraction scheme of chiral perturbatheory) [12,13]. To obtain the final renormalized expression we perform additional finite subtractions. Fpurpose we expand the coefficients and integrals in Eq. (3) in powers ofp2 − m2 and /p − m, both countingas O(Q). The integralsIρ(0) and IN(0) do not depend onp2; IρN(0,−p) is an analytic function ofp2 inthe vicinity of the pointp2 = m2. Hence the expansion of Eq. (3) in powers ofp2 − m2 contains only integepowers. The final renormalized expression forΣ1 is obtained by subtracting from Eq. (3) the terms proportiona(p2 −m2)0, (p2 −m2), /p −m, (p2 −m2)(/p −m), and(p2 −m2)2.3 As a result of this subtraction we obtain ththe renormalized self energyΣR

1 starts as(p2−m2)3 in agreement with power counting asn → 4. All countertermscorresponding to the above subtractions are generated by expanding bare quantities of the Lagrangian inrenormalized ones.

As the second example we consider the vertex diagram of Fig. 2. Applying Feynman rules, the corresexpression reads

(5)V a = ig2gA

2Fτa

∫dnk

(2π)n(/q − /k)γ5

1

/p + /k − m + iεγ µ

gµν − kµkν

M2ρ

k2 − M2ρ + iε

(2q − k)ν

(q − k)2 − M2 + iε.

Power counting suggests that this diagram is of orderQ3. For the sake of simplicity, we only consider Eq. (evaluated between on-mass-shell nucleons [p2 = m2 = (p + q)2],4

3 We do not quote the specific values of the subtraction constants, because they are not relevant for our discussion.4 Recall thatu(p + q)γ5u(p) counts asO(Q) [13].

14 T. Fuchs et al. / Physics Letters B 575 (2003) 11–17

ls to

by firstl to

Fig. 2. One-loop contribution to theπNN vertex including an internalρ meson.

u(p + q)V au(p) = g2gA

m

F

{AIρπ(0,−q)+ B

[Iρπ (0,−q)− Iρπ (0,0)

] + CJ(112|n+ 2)

(6)+ DJ(121|n+ 2)+ EIρπN (0,−q,p)+ FIρ(0)}u(p + q)γ5τ

au(p),

where the coefficientsA, . . . ,F are given by

A = −1

2

(5+ q2 −M2

M2ρ

), B = M2

ρ − M2

2q2

(1+ q2 − M2

M2ρ

), C = 8π

(2m2 − q2),

D = −8πq2, E = −2q2, F = 1

M2ρ

,

and the integrals read

Iρπ (0,p) = i

∫dnk

(2π)n

1

(k2 − M2ρ + iε)[(k + p)2 − M2 + iε] ,

(7)IρπN(0,−q,p) = i

∫dnk

(2π)n

1

(k2 − M2ρ + iε)[(k − q)2 − M2 + iε][(k + p)2 − m2 + iε] .

Moreover, we have introduced the auxiliary integral

(8)J (abc|d)= i

∫ddk

(2π)d

1

(k2 −M2ρ + iε)a[(k − q)2 − M2 + iε]b(k2 + 2p · k + iε)c

,

which, with shifted space–time dimensionn + 2, contributes to Eq. (6) as a result of reducing tensor integrascalar ones. An explicit calculation yields

Iρπ (0,−q)= −1

(4π)n2

(M2

ρ

) n2−2

∞∑l=0

1

l!(

q2

M2ρ

)l*(2− n/2+ l)*(1+ l)*(1+ l)

*(2+ 2l)

× F

(1+ l,2− n

2+ l;2+ 2l;1− M2

M2ρ

)(9)= − 1

(4π)n2

(M2

ρ

) n2−2

*(2 − n/2) − 1

16π2− 1

8π2

M2

M2ρ

ln

(M

Mρ

)− q2

32π2M2ρ

+O(Q4),

whereF(a, b; c; z) is the hypergeometric function [19] andQ stands for eitherq or M.Again, we find that the unrenormalized diagram violates the power counting. We renormalize Eq. (6)

subtracting all ultraviolet divergences using theMS scheme which amounts to dropping all terms proportiona

T. Fuchs et al. / Physics Letters B 575 (2003) 11–17 15

nts andtegerting.cting all

e., is of

ponding

Fig. 3. Pion self energy diagram withρ meson dressing.

λ (see Eqs. (10) and (12) below). To determine the additional finite subtractions, we expand all coefficieintegrals in powers ofM2 andq2. The integrals contain non-analytic parts which are proportional to non-inpowers ofM2 and/orq2 for non-integern. These non-analytic contributions separately satisfy the power counWe only expand the analytic parts and obtain the final renormalized expression of the diagram by subtraterms of the above expansion which are of orderQ2 or less. The terms which need to be subtracted read

u(p + q)V asubtru(p)

(10)

= g2gA

m

2F

{5(M2

ρ)n2−2

(4π)n2

*(2− n/2) + 5

16π2 − 1

32π2

+ 32πm2J0(112|n+ 2)+ 2

M2ρ

Iρ(0) − 2q2 − M2

M2ρ

λ

}u(p + q)γ5τ

au(p),

where

(11)J0(abc|d)= i

∫ddk

(2π)d

1

(k2 − M2ρ + iε)a(k2 + iε)b(k2 + 2p · k + iε)c

,

and

(12)λ = mn−4

16π2

{1

n − 4− 1

2

[ln(4π) + *′(1)+ 1

]}.

Taking into account that

J (112|n+ 2)− J0(112|n+ 2) ∼ Q2,

it is now straightforward to check that the difference of Eq. (6) and Eq. (10) satisfies the power counting, i.orderQ3 asn → 4.

As a final example, we discuss the one-loop diagram of the pion self energy given in Fig. 3. The corresexpression reads

(13)−iΣabπ (p) = −iΣπ(p)δab = −g2εacdεbcd

∫dnk

(2π)n

(2p + k)µ(2p + k)ν(gµν − kµkν

M2ρ

)[(k + p)2 − M2 + iε](k2 − M2

ρ + iε),

from which we obtain

(14)Σπ(p) = −2g2

{(1+ p2 − M2

M2ρ

)Iρ(0) +

[2M2 + 2p2 − M2

ρ − (p2 − M2)2

M2ρ

]Iρπ (0,p) − Iπ (0)

},

where

(15)Iπ (0) = −M2(M2ρ)

n2−2*(2− n/2)

(4π)n2

− M2

16π2 + M2

8π2 ln

(M

Mρ

),

andIρπ (0,p) is given in Eq. (9).

16 T. Fuchs et al. / Physics Letters B 575 (2003) 11–17

rmsntifying

thealogous

same.ing afteralytic inn.

articlesuitableulationwhere ant with

acknowl-

To renormalize the pion self energy we first apply theMS scheme. The additional finite subtraction counterteare obtained by expanding the coefficients and the analytic parts of the integrals in Eq. (14) and idethose terms which are of lower order than suggested by power counting, i.e., orderQ4. Note that the last termin Eq. (15), which is non-analytic inM, will give a contribution to Eq. (14) which, if taken separately, violatespower counting. It cannot be removed by counterterms in the Lagrangian, but exactly cancels with an ancontribution coming from theIρπ (0,p) term in Eq. (14). We arrive at the following renormalized expression:

ΣRπ (p) = −2g2

{(2M2 + 2p2 − M2

ρ

)[Iρπ (0,p) + 1

16π2+ (M2

ρ)n2−2*(2− n/2)

(4π)n/2

]

(16)− (p2 − M2)2

M2ρ

[Iρπ (0,p) − 2λ

] − p2

32π2 − M2

8π2 ln

(M

Mρ

)}.

Using Eq. (9) we see that Eq. (16) satisfies the power counting, i.e., is of orderQ4.Two- (and multi-) loop diagrams have a more complicated structure, but the outcome remains the

Those terms which are non-analytic in small expansion parameters satisfy the systematic power countsubtracting the one-loop-order sub-diagrams. The contributions which violate the power counting are ansmall expansion parameters and are subtracted by a finite number of local counterterms in the Lagrangia

4. Summary and conclusions

We have demonstrated that the inclusion of explicit degrees of freedom corresponding to (axial) vector pin manifestly Lorentz-invariant baryon chiral perturbation theory does not violate the power counting if a srenormalization condition is used. As an important test of our method it is now necessary to apply a full calcto physical processes such as, e.g., the determination of the nucleon electromagnetic form factors [20],one-loop calculation in ordinary baryon chiral perturbation theory does not show a satisfactory agreemedata beyond very small values ofQ2 ≈ 0.1 GeV2 [21,22].

Acknowledgements

The work of T.F. and S.S. was supported by the Deutsche Forschungsgemeinschaft (SFB 443). J.G.edges the support of the Alexander von Humboldt Foundation.

References

[1] J.S. Schwinger, Phys. Lett. B 24 (1967) 473.[2] J. Wess, B. Zumino, Phys. Rev. 163 (1967) 1727.[3] S. Weinberg, Phys. Rev. 166 (1968) 1568.[4] M. Bando, T. Kugo, K. Yamawaki, Phys. Rep. 164 (1988) 217.[5] U.-G. Meißner, Phys. Rep. 161 (1988) 213.[6] M. Harada, K. Yamawaki, Phys. Rep. 381 (2003) 1.[7] G. Ecker, J. Gasser, H. Leutwyler, A. Pich, E. de Rafael, Phys. Lett. B 223 (1989) 425.[8] G. Ecker, J. Gasser, A. Pich, E. de Rafael, Nucl. Phys. B 321 (1989) 311.[9] B. Borasoy, U.-G. Meißner, Int. J. Mod. Phys. A 11 (1996) 5183.

[10] M.C. Birse, Z. Phys. A 355 (1996) 231.[11] S. Weinberg, Physica A 96 (1979) 327.[12] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142.

T. Fuchs et al. / Physics Letters B 575 (2003) 11–17 17

[13] J. Gasser, M.E. Sainio, A. Svarc, Nucl. Phys. B 307 (1988) 779.[14] U.-G. Meißner, PiN Newslett. 16 (2002) 1.[15] T. Fuchs, J. Gegelia, G. Japaridze, S. Scherer, Phys. Rev. D 68 (2003) 056005.[16] J. Gegelia, G. Japaridze, Phys. Rev. D 60 (1999) 114038.[17] J. Gegelia, G. Japaridze, X.Q. Wang, J. Phys. G 29 (2003) 2303.[18] S. Weinberg, Nucl. Phys. B 363 (1991) 3.[19] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1972.[20] M.R. Schindler, J. Gegelia, S. Scherer, in preparation.[21] B. Kubis, U.-G. Meißner, Nucl. Phys. A 679 (2001) 698.[22] T. Fuchs, J. Gegelia, S. Scherer, nucl-th/0305070.