A Comment on the Renormalization of the Nonlinear Sigma Model

IFUM-FT/883

arXiv:hep-th/0701197v1 22 Jan 2007

A Comment on the Renormalization of the Nonlinear Sigma Model

D. Bettinelli1, R. Ferrari2, A. Quadri3 Dip. di Fisica, Università degli Studi di Milano and INFN, Sez. di Milano via Celoria 16, I-20133 Milano, Italy

Abstract We consider the recently proposed renormalization procedure for the nonlinear sigma model, consisting in the recursive subtraction of the divergences in a symmetric fashion. We compare this subtraction with the conventional procedure in power counting renormalizable (PCR) theories. We argue that symmetric subtraction in the nonlinear sigma model does not follow the lore by which nonrenormalizable theories require an infinite number of parameter fixings. Our conclusion is that only two parameters can be consistently used as physical constants.

1

e-mail: [email protected] e-mail: [email protected] 3 e-mail: [email protected] 2

1

1

Introduction

The matter needs some semantics, thus we use the extended notation of “power counting renormalizable theory” (PCR) when we deal with conventional renormalizable theories. We use the notion of “symmetric subtracted theory” (SySub) when the perturbation series can be made finite by the subtraction of infinities in a symmetric and local fashion and is therefore renormalizable in the ‘modern’ sense of [1]. An example of this latter situation is provided by the nonlinear sigma model in a recent formulation in terms of flat connection [2]-[4]. Just this theory is our paradigm for the discussion about the number of parameters that have to be fixed in order to define the subtractions in the perturbative series. The issue of the number of physical parameters in a theory which is not renormalizable by power-counting has been discussed several times in the recent literature. In [5] it has been proposed to introduce a framework for reducing [6] the infinite number of free parameters to a smaller, eventually finite, one. A similar strategy has been advocated in [7] in the context of Wilson’s approach to renormalization [8]. In this paper we argue that the lore, by which an infinite number of experiments is required in order to fix the counterterms for a nonrenormalizable theory, stems from an inappropriate use of the point of view of the algebraic renormalization [9]-[12] to theories that cannot be treated according to such a procedure. In the case of the nonlinear sigma model the theory is defined through the effective action which has to obey a nonlinear local functional equation. At the one loop level the counterterms obey a linearized form of the same equation. These counterterms have a particular feature: they are not present in the classical action. Some of them do not obey the nonlinear defining functional equation. Others modify in a substantial way the unperturbed space of states. Finally there are some that could be introduced in the action at the tree level. In any case, however, this modification of the classical action would produce a new set of one loop counterterms. Thus the procedure of assigning free parameters to the counterterms is not viable. We discuss also the possibility of assigning free parameters to the counterterms at the one loop level. We argue that this strategy is not sustainable from the physical point of view, since parameters should enter in the classical action. A particular case of the parameters appearing only in the radiative corrections is an extra mass scale introduced in order to perform dimensional subtraction. We argue that this parameter has the very important role of fixing the scale of the radiative corrections. One can formulate the model in 2

such a way that the dimensional subtraction scale appears as a front factors of the whole set of Feynman rules. The final consequence of this physical requirement is that the nonlinear sigma model depends on only two parameters, e.g. the v.e.v. of the spontaneous breakdown and the dimensional subtraction scale. This choice of the independent parameters turns out to be useful for investigations where the rôle of the subtraction is not essential as long it is symmetric. An example of this situation is provided by the study of the large mass limit in the linear sigma model recently performed in [13]. The paper is devoted to a detailed illustration of the above mentioned facts and it is written in the spirit of a novel view on those nonrenormalizable theories that can be consistently subtracted (i.e. symmetrically and locally). The discussion is done at the one loop level. We discuss briefly the general case of n-loops, in particular we guess the form of the equation obeyed in the presence of counterterms.

2

The Nonlinear Sigma Model

The D-dimensional classical action of the nonlinear sigma model in the flat connection formalism [2] is 4 Z 2 Z m2D (0) D µ µ Γ = d x Fa − Ja + dD x K0 φ0 (1) 8 D

where mD = m 2 −1 . The flat connection is i 1 F µ = Faµ τa = Ω∂µ Ω† 2 g 1 Ω= (φ0 + igτa φa ) (2) mD where Jµa is the background connection and K0 is the source of the constraint φ0 of the nonlinear sigma model φ20 + g 2 φ2j = m2D .

(3)

Γ(0) obeys a D-dimensional local functional equation associated to the local chiral transformations induced by left multiplication on Ω by SU(2) matrices. This equation is required to be valid for the effective action on the basis of a path integral formulation of the model − ∂µ

δΓ 1 δΓ 1 δΓ δΓ 1 µ δΓ + g 2 φ a K0 + = 0. µ − gǫabc Jb µ + g ǫabc φc δJa δJc 2 δφb 2 2 δK0 δφa 2

(4)

m Here the external current J~µ of Ref. [2] has been rescaled by a factor − 4D and a harmless J 2 has been introduced in the action. 4

3

The equation is local (no x-integration). The spontaneous breakdown of the global chiral symmetry is fixed by the boundary condition δΓ = mD . (5) δK0 φ= ~ J~µ =K0 =0

It will be required that these equations ((4) and (5)) remain valid also for the subtracted amplitudes (symmetric subtraction). The non linearity of the equation (4) is responsible for many peculiar facts. δΓ In particular by eq.(5) δK is invertible as a formal power series. Therefore 0 ~ fields (descendants) can be by using eq.(4) all amplitudes involving the φ derived from those of F~µ and φ0 (ancestors), i.e. the functional derivatives with respect to J~µ and K0 (hierarchy). The tree level amplitudes are fixed by the conditions m2D δ 2 Γ(0) = gµν δab δJaµ (x)δJbν (y) 4 δ 2 Γ(0) =0 δK0 (x)δK0 (y) δ 2 Γ(0) = 0. δK0 (x)δJbν (y)

(6)

The dependence of the solution from the parameter g is somehow pecu~ mD ] of equation (4) with the ~ K0 , φ, liar. Given an unsubtracted solution Γ[J, boundary conditions (5) and (6), one can check that ~ g mD ] Γ[g −1J~, g −1K0 , φ,

(7)

obeys the same equations with g = 1. Thus g can be removed by a redefinition of the mass scale parameter mD → g mD (together with J~ → g −1 J~ and K0 → g −1 K0 ). However the situation changes if one wants to define the theory at D = 4. Subtraction of poles is needed and, together with this, a scale parameter in the definition of the Feynman amplitudes is necessary. At one loop level the dependence of the subtracted amplitudes from ln m (in D = 4) doesn’t allow the complete removal of g. Thus, at least at the one loop level, the introduction of g is equivalent to use an extra mass scale in the dimensional subtraction and accounts for variants of the minimal subtraction. There is another interesting rescaling strategy, i.e. consider ~ mD ]. ~ K0 , g −1φ, Γ[g −1J, 4

(8)

This vertex functional satisfies the eq. (4) with g = 1 and eq. (5) unchanged. But eq. (6) becomes 2 (D−4) δ 2 Γ(0) m m2D m gµν δab (9) = 2 gµν δab = µ ν δJa δJb 4g g 4 i.e., again, we have a new mass parameter m v≡ . g

(10)

The discussion on the rôle of the parameter g will be resumed and expanded in Sec. 6.

3

Subtractions at D = 4 (one loop)

Subtractions at D = 4 are performed in dimensional regularization. At one loop level the counterterms obey a linearized form of the above local equation h (0) 1 δΓ(0) δ 1 δ δ b(1) ) = 1 δΓ S0 (Γ + + g ǫabc φc 2 δφa δK0 2 δK0 δφa 2 δφb i δ b(1) δ = 0. (11) −∂ µ µ − gǫabc Jbµ µ Γ δJa δJc

It is easy to trace in eq. (11) the transformations induced through J and φa . Further properties can be derived by introducing the Grassmann parameter ωa and the nilpotent operator [3] Z Z δ h 1 g D ωa φ0 + ǫajk φj ωk s ≡ d x ωa S0 = dD x 2 2 δφa i (0) 1 δΓ δ δ + ωa (12) + ∂ µ ωa + gǫaij Jiµ ωj 2 δφa δK0 δJaµ

with

g s ωa = − ǫajk ωj ωk . 2

(13)

We consider the Legendre transform Ka ≡ −

δ (0) δ K0 Γ =− S0 + g 2 φ a δφa δφa φ0

where S0 = Γ(0) K0 =0 . One gets

g g2 s Ka′ = − ǫa′ ak ωk Ka + ωa′ K0 2 2 5

(14)

(15)

and 1 s K 0 = − ω a Ka . 2

(16)

Then K 0 ≡ K 0 φ 0 + Ka φ a K0 δ S0 + g 2 φ a φ a = K0 φ 0 − φ a δφa φ0 2 m K0 δ = D − φa S0 φ0 δφa

(17)

is a local solution of s K 0 = 0.

(18)

In terms of the background connection Jaµ and of the flat connection Faµ =

2 φ0 ∂ µ φa − ∂ µ φ0 φa + gǫabc ∂ µ φb φc 2 mD

(19)

the invariant solutions of the linearized functional equation which enter at the one loop level read [3] Z

h i h i dD x Dµ (F − J)ν D µ (F − J)ν , a a Z i i h h I2 = dD x Dµ (F − J)µ Dν (F − J)ν , a a Z h i I3 = dD x ǫabc Dµ (F − J)ν Fbµ − Jbµ Fcν − Jcν , a Z m2 K 2 δS 0 D 0 I4 = dD x − φa , φ0 δφa Z m2 K 2 δS0 µ D 0 I5 = dD x − φa Fb − Jbµ , φ0 δφa Z 2 2 Fbν − Jbν , I6 = dD x Faµ − Jaµ Z D µ µ ν ν I7 = d x Fa − Ja Fa − Ja Fbµ − Jbµ Fbν − Jbν , (20) I1 =

where Dµ denotes the covariant derivative w.r.t Fµa : Dabµ = δab ∂µ + gǫacb Fcµ . 6

(21)

By dimensional arguments one expects that at one loop the counterterms (the 1/(D − 4) pole parts) are linear combinations of I1 . . . I7 . In Ref. [3] the linear combination is explicitly evaluated. On these grounds other solutions of eq.(11) are excluded, e.g. Z dD xK 0 . (22)

3.1

D = 4 at higher loops

It is worth mentioning here that at higher loops the counterterms obey a more complex equation. Some explicit calculations at two loops [4] and formal arguments based on quantum action principle suggest that the equation is h 1 δΓ(0) δ 1 δΓ(0) δ 1 δ (n) b S0 (Γ ) = + + g ǫabc φc 2 δφa δK0 2 δK0 δφa 2 δφb n−1 i b(n−j) b(j) δ Γ 1 X δΓ δ b(n) δ =− , −∂ µ µ − gǫabc Jbµ µ Γ δJa δJc 2 j=1 δK0 δφa

(23)

provided that symmetric subtraction is performed correctly. By standard arguments ([14], [15]) one can show the validity of the consistency condition ! Z n−1 b (j) b (n−j) X δ Γ δ Γ s dD x ωa =0 (24) δK δφ 0 a j=1 under the assumption that eq.(23) is recursively fulfilled up to order n − 1. For the discussion presented in the next sections it is worth to outline the arguments that lead to eq. (23). Consider the 1PI generating functional b(j) have been introduced up to n − 1 loops. Then at where counterterms Γ n-loops poles in D − 4 are present and moreover we expect a violation of eq. (4) (n) δΓ(n) 1 δΓ(n) µ δΓ µ − gǫabc Jb µ + g ǫabc φc δJa δJc 2 δφb n−1 (0) (n) (0) (n) 1 δΓ δΓ 1 X δΓ(n−j) δΓ(j) 1 δΓ δΓ + + + 2 δK0 δφa 2 δφa δK0 2 j=1 δK0 δφa

−∂ µ

= ∆(n) · Γ ,

(25)

where ∆(n) · is the insertion of the local operator ∆(n) . Since the bilinear terms have no poles in D − 4, the procedure of minimal subtraction yields 7

n-loop counterterms that obey a non homogeneous linearized equation. The violation term in eq. (23) has this particular form by the following argument. Consider the formal perturbative expansion of the functional generating the b(j) have been introconnected Feynman amplitudes where counterterms Γ duced X ~ K0 , K) ~ = exp i Γ(0) + b(j) ZR (J, Γ INT δ exp

1 2

j=1

Z

φa =−i δK

dD x dD y Kb (x)DF (x − y)Kb(y).

a

(26)

By standard formal argument the equation (4) can be written as an insertion of the relevant derivatives of the classical vertex functional and of the counterterms δ δ 1 δ 1 δ δ 1 ∂ µ µ + gǫabc Jbµ µ − g 2 K0 ZR = + Ka + gǫabc Kb δJa δJc 2 δKa 2 δK0 2 δKc b b δΓ b b b δΓ 1 δΓ 1 2 1 µ δΓ µ δΓ − − g K0 φ a · Z R + gǫabc Jb µ − g ǫabc φc i ∂ δJaµ δJc 2 δφb 2 δK0 δφa 2 = 0, (27) b is a shorthand notation for where Γ

b = Γ(0) + Γ

X j=1

b (j) . Γ

(28)

The grading in ~ gives very useful relations (23). In particular, if divergences are subtracted up to n − 1-loops the violation of the functional equation (25) at n loops is due to the fact that the relevant counterterms are absent in eq. (27). Eq. (27) indicates that the non homogeneous part is n−1 b (j) b (n−j) δΓ 1 X δΓ − 2 j=1 δK0 δφa

(29)

and therefore the insertion in eq. (25) is (n)

∆

n−1 b (j) b (n−j) 1 X δΓ δΓ =− . 2 j=1 δK0 δφa

(30)

This particular form of the breaking term allows a recursive symmetric subtraction by dimensional renormalization. If this strategy fails, a finite 8

renormalization has to be performed in order to reestablish the functional equation. There is a further important relation among 1PI vertex functions that can be derived from eq. (23). Let us define by Γ(n,k) the n-loop 1PI functional where the sum of the order of the counterterms is a fixed k ≤ n. Then the vertex function at n-loop is X Γ(n) = Γ(n,j) . (31) j=0,n

The following equation (n > 0) can be proved by induction ! (0) (0) δ δ 1 δΓ δ 1 δΓ δ δ 1 −∂ µ µ − gǫabc Jbµ µ + g ǫabc φc Γ(n,k) + + δJa δJc 2 δφb 2 δK0 δφa 2 δφa δK0 n−1 1X + 2 n′ =1

j≤min(k,n′ )

X

′

j≥max(0,k−n+n′ )

′

δΓ(n ,j) δΓ(n−n ,k−j) =0 δK0 δφa (32)

together with the consistency condition  Z j≤min(k,n′ ) n−1 X X D s d xωa (x) 

n′ =1 j≥max(0,k−n+n′ )

4

(n′ ,j)

(n−n′ ,k−j)

δΓ δΓ δK0 δφa



 = 0.

(33)

Parameters fixing

Minimal subtraction is of course a very interesting option in order to make finite the perturbative series. The conjecture that this subtraction algorithm is symmetric (i.e. eq. (4) is stable) is supported by some general arguments (given in Sec. 3) and by an explicit example in Ref. [4]. Thus, at the moment that this theory is consistently defined, it can be tested by experiments. A frequent objection to the present proposal of making finite a nonrenormalizable theory is that one needs seven parameter fixing appropriate measures in order to evaluate the coefficients of I1 . . . I7 . This objection is legitimate if the above mentioned invariants are action-like. As one should do in power counting renormalizable theories, according to algebraic renormalization. Here the situation is more involved. This is evident if we paraphrase 9

the problem in the following way. Can we introduce in the unperturbed action the seven invariants with arbitrary coefficients and treat them as bona fide interaction terms intervening in the loop expansion as the original one provided in Γ(0) of eq. (1)? The answer to this question is in general negative. If one allows this modification of the unperturbed action, the one loop corrections will be modified by extra terms generated by the newly introduced Feynman rules, thus bringing to a never ending story. In particular the introduction at tree level of the vertices described by the invariants in eq.(20) implies new Feynman rules which invalidate the weak power-counting [3]. The superficial degree of divergence of the ancestor amplitudes in not any more given by δ = (D − 2)n + 2 − NJ − 2NK0

(34)

(where NJ and NK0 are the numbers of insertions of flat connections Faµ and constraints φ0 ). As a direct consequence of the violation of the weak powercounting, already at one loop the number of divergent ancestor amplitudes is infinite. A closer look to I1 . . . I7 shows that there are also other reasons that forbid the use of some of these invariants as unperturbed action terms. I1 , I2 can be introduced into Γ(0) without breaking the eq. (4). However they modify the spectrum of the unperturbed states (by introducing negative norm states) through kinetic terms with four derivatives. I4 , I5 cannot be introduced into Γ(0) because they violate eq. (4).

5

Finite subtractions

After we excluded the possibility of introducing in the classical action the invariants I1 . . . I7 , there is still the possibility to use them for a finite, in principle arbitrary, renormalization. I.e. in the book keeping of the Feynman rules one could enter new terms X Z ~ λj dD x Ij (x) , (35) j

where we have explicitly exhibited the ~ factor in order to remind that these vertexes are of first order in ~ expansion. λj are arbitrary real parameters. More explicitly we can tell the story in the following way. The subtraction of the poles in D − 4 requires a series of counterterms of the form (35) where the coefficients carry the pole factor 1/(D − 4). Then it seems reasonable to use these extra degrees of freedom as free parameters. 10

In the PCR case the fixing of the finite parts of the symmetric counterterms can be seen as a way to introduce the renormalization by a reset of the parameters entering into the classical action. The situation is clearly different in the present case, since the invariants I1 , . . . , I7 are not action-like and therefore the additional parameters λj can be introduced only as quantum corrections. The meaning of this latter procedure, outside an effective field theory approach [16], seems to us rather unclear from the physical point of view, since independent parameters are used in the radiative corrections. The alternative approach (which we favour) is to assume that all SySub (not only PCR) theories should obey the principle ruling PCR models, namely that parameters have to be introduced ab initio in the classical action.

6

A proposal: Ockham’s razor

The above discussion illustrate the fact that we face an antinomy. From a mathematical point of view, finite subtractions as in eq. (35) are allowed and yield the most general solution to the subtraction procedure. From a physical point of view, free parameters as λj cannot be introduced in the radiative corrections. We avoid this antinomy, if we assign a preeminent rôle to the pure pole subtraction, i.e. only minimal subtraction is used in order to make the theory finite at D = 4. However, even with this clear cut strategy, still there is some freedom left connected to the presence of g or equivalently to the use of a second scale parameter in the Feynman rules in dimensional renormalization. Here we would like to give a formulation of this choice that has some appeal. We have to extract a finite part from a generic amplitude in D dimensions involving n external currents J Γ[J1 · · · Jn |D]

(36)

This can be done by using the normalized function m2 1 Γ[J1 · · · Jn |D] = (D−4) Γ[J1 · · · Jn |D] 2 mD m

(37)

and then by subtracting out the pole parts in D = 4. For example, the single pole part in Γ[J1 · · · Jn , |D] is removed by the counterterm mechanism Γ[J1 · · · Jn , |D] −

m2D ′ lim (D ′ − 4)m−2 D ′ Γ[J1 · · · Jn , |D ] . (D − 4) D→4 11

(38)

The normalization used in eq. (37) is needed in order to produce the correct dimensions of the counterterms in eq. (38). Similarly one proceeds with K0 . The normalized function is n mD−4 Γ[K01 · · · K0n |D] = m(1− 2 )(D−4) Γ[K01 · · · K0n |D]. (39) −2) n( D m 2 Eq. (37) show that the parameter g can be removed only in unsubtracted amplitudes (i.e. at D 6= 4). At the one loop level the replacement mD → 2 g mD (m → mg D−2 )leads to a ln g dependence of the amplitude in D = 4. In fact eq. (37) becomes (4−D) 2 Γ[J1 · · · Jn |D]. (40) m g D−2

Thus the minimal subtraction introduces in this case a new mass scale µ ≡ m g. The formulation with two parameters takes a particular elegant form if we suppress g in favor of a second mass scale and moreover we assign to K0 a dimension that is D-independent; i.e. in a way that the normalization factor for the subtraction of the poles is identical both for J~µ and K0 . To achieve this normalization we start from the transformation in eqs. (8) and (10) and perform the further transformation ~ mD ]. ~ v mD K0 , mD φ, Γ[J, (41) m2 v The functional equation (4) is replaced by 1 1 δΓ δΓ µ δΓ − m(D−4) φa K0 µ − ǫabc Jb µ + ǫabc φc δJa δJc 2 δφb 2 1 1 δΓ δΓ + = 0, 2 m(D−4) δK0 δφa

−∂ µ

(42)

the constraint condition (3) replaced by φ20 + φ2j = v 2 . and eq. (5) by

δΓ = v mD−4 , δK0 φ= ~ J~µ =K0 =0

(43)

(44)

while eq. (9) is unchanged 5 .

v 2 (D−4) δ 2 Γ(0) m gµν δab . = δJaµ δJbν 4

(45)

~→vφ ~ and K0 → 1/vK0 . In fact the Note that v cannot be removed by a rescaling φ dependence on v remains in eq. (45) 5

12

This amounts to formally perform the path-integral according to (DΩ denotes the invariant Haar measure over SU(2)) ~ K0 , K] ~ Z[J, 2 Z Z v D (D−4) µ µ 2 d x = DΩ exp im (Fa − Ja ) + K0 φ0 + Ka φa . (46) 8 By this choice the dimensions of J~µ is one and K0 is three. The evaluation of the counterterms is then the same (independently from the number of J~µ and of K0 ), via simple pole subtraction of the normalized functions as in eq. (37) (D−4) 1 (47) Γ[J1 · · · Jn K01 · · · K0n′ |D]. m Then the full set of Feynman rules is ! Z 2 X v 2 b = m(D−4) dD x (Faµ − Jaµ ) + K0 φ0 + M(j) , Γ 8 j=1

(48)

where the M(j) are the local counterterms containing the pole parts in D = 4. The finite parts of the subtractions is governed by the sole front factor m(D−4) . The resulting amplitudes depend on the parameters v and m. The last one is not present in the classical action: it sneaks in as a scale of the radiative corrections. A similar mechanism has a renowned antecedent in the theory of Lamb shift [17], where the radiative corrections due to the excited state transitions need a ultraviolet cut-off which is not present at the lowest level of the theory of the Hydrogen atom. A comment is in order here. In PCR theories the free parameters in the classical action can be fixed by a set of normalization conditions at a given mass scale m. Moreover, a shift in m is reabsorbed by a shift in the same free parameters entering into the classical action (renormalization group). On the contrary in the NLSM a shift in m cannot be compensated by a shift in v. Therefore m has to be treated as a second independent free parameter (in addition to v) to be determined through the fit with the experimental data.

7

Conclusions

From the mathematical point of view Symmetrical Subtraction of infinities in the nonlinear sigma model is possible by using minimal subtraction in 13

dimensional renormalization (this is a conjecture based on non trivial examples and on formal arguments). The resulting theory depends on m and g. However at each order of the perturbative series one can introduce finite renormalizations by using the appropriate local solutions of eq. (23). For instance at one loop level the equation takes the form exhibited in (35). In this general scheme the restoring of the functional equation (4) becomes a very complex procedure: one needs to solve the non homogeneous linearized equation as in eq. (23) but with a more complex non homogeneous term. From the physical point of view a theory is acceptable if all parameters appear in the classical action. By following strictly this criterion, the only admissible theory is the one where there is only one free parameter. In fact the extra parameter g can be removed in the unsubtracted amplitudes and it appears only in the procedure of the pole removal i.e. in the quantum corrections. The antinomy described in Sections 4 and 5 is solved only by strict minimal subtraction in the nonlinear sigma model with v as unique parameter. However this criterion can be relaxed by allowing a second mass parameter which enters as a scale of the radiative corrections. We formulated the symmetrically subtracted nonlinear sigma model in such a way that the second parameter enters as a common front factor of the whole Feynman rules (counterterms included).

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