Physics Letters B 720 (2013) Contents lists available at SciVerse ScienceDirect. Physics Letters B.

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1 Physics etters B ) Contents lists available at SciVerse ScienceDirect Physics etters B Infrared divergences, mass shell singularities and gauge deendence of the dynamical fermion mass Ashok K. Das a,b,,j.frenkel c, C. Schubert d a Deartment of Physics and Astronomy, University of Rochester, Rochester, NY , USA b Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta , India c Instituto de Física, Universidade de São Paulo, , São Paulo, SP, Brazil d Institute for Physics, Michoacan University, C.P , Morelia, Michoacan, Mexico article info abstract Article history: Received 10 December 01 Received in revised form 0 February 013 Acceted 1 February 013 Availableonline6February013 Editor: M. Cvetič We study the behavior of the dynamical fermion mass when infrared divergences and mass shell singularities are resent in a gauge theory. In articular, in the massive Schwinger model in covariant gauges we find that the ole of the fermion roagator is divergent and gauge deendent at one loo, but the leading singularities cancel in the quenched rainbow aroximation. On the other hand, in hysical gauges, we find that the dynamical fermion mass is finite and gauge indeendent at least u to one loo. 013 Elsevier B.V. All rights reserved. The ole of the roagator for a article defines the mass of the article and, in a gauge theory, the ole is exected to be gauge indeendent 1,]. Intuitively it is clear that the hysical mass of a article should be gauge indeendent and quantitatively it can be argued as follows. et us consider a fermion of tree level mass m interactingwithahotonfieldinacovariantgauge.thenthehysical mass m of the fermion is determined from the vanishing of the comlete two oint function S 1 = / m Σ,m) = 0. Here Σ,m) denotes the fermion self-energy to all orders. At one loo, for examle, the fermion self-energy see Fig. 1) in D dimensions leads to the equation Σ 1) = e / m)f 1 + F / m) ), 1) ξ where ξ denotes the covariant) gauge fixing arameter and F 1,m) = 1 d D k 1 1 ) D k ) /k + / m /k, F,m) = 1 d D k 1 ) D k ) /k 1 /k + / m. ) Since to one loo order m 1) = m + Oe ) and the right hand side of 1) is already of order e, we can let m m 1) in the terms * Corresonding author at: Deartment of Physics and Astronomy, University of Rochester, Rochester, NY , USA. addresses: das@as.rochester.edu A.K. Das), jfrenkel@fma.if.us.br J. Frenkel), schubert@ifm.umich.mx C. Schubert). Fig. 1. Fermion self-energy in QED at one loo. / m) on the right hand side which shows that the right hand side vanishes at the ole / = m 1) if the functions F 1, F are well behaved. Furthermore, if we can Taylor exand the self-energy around the ole to write Σ 1),m) = m 1) ) m + / m 1) ) Σ 1) ) 1) /=m +, 3) onlythefirsttermsurvivesattheole.asaresult,attheole1) leads to m1) ξ = 0 showing that the ole of the roagator mass of the article) is gauge indeendent. Gauge indeendence of the ole at higher loos is best studied through an analysis of the Nielsen identity for the two oint function which we will discuss later 3 5]. It is clear from ) that the functions F 1, F become infrared singular for sace time dimensions D 3 so that the right hand side of 1) may not vanish in general. In addition, as we will see in lower dimensional models, mass shell singularities develo in the fermion self-energy so that a Taylor exansion as in 3)) is not ossible and the gauge indeendence of the ole of the roagator cannot be guaranteed. These are two ossible difficulties that can invalidate the formal argument of gauge indeendence /$ see front matter 013 Elsevier B.V. All rights reserved. htt://dx.doi.org/ /j.hysletb

2 A.K. Das et al. / Physics etters B ) given above and when infrared divergences and/or mass singularities are resent, the question of gauge indeendence of the ole has to be examined anew. This is imortant since the effective action and its imaginary art) for fermions interacting with a gauge field background deends on the fermion mass 6] and being a hysical quantity the imaginary art is related to the vacuum decay rate), it cannot ossibly be gauge deendent. Infrared singularities are well known to be a roblem in lower dimensions 7,8], but even in 4-dimensional QCD, they can arise in the fermion self-energy) at two loos and beyond because of gluon loos 9]. Of course, intuitively it is clear that a strong infrared divergence signifies ossible confinement of the fermion in the theory without any hysical fermion asymtotic states. As we know this is true even in Abelian theories in lower dimensions where the strong infrared behavior of the Coulomb force leads to confinement of fermions 7,10 13].) Nonetheless one can calculate the fermion mass for the erturbative asymtotic states as is done in the case of current quark masses which are then used in various calculations and it is worth investigating whether such a calculation can lead to a gauge indeendent result. For these reasons we have chosen to study this question systematically. We find that in covariant gauges infrared divergences as well as mass shell singularities develo in the fermion selfenergy at one loo when D 3 invalidating the standard arguments for gauge indeendence of the ole of the roagator. However, we show that the leading divergences cancel when the fermion self-energy is summed to all orders in the quenched rainbow aroximation 14,15]. This signals the fact that the infrared divergences and mass shell singularities may not be resent in a hysical gauge such as the axial and Coulomb gauges. We carry out such a calculation for the 1 + 1)-dimensional massive QED and show exlicitly that there are no infrared divergences in the self-energy u to two loos. We also find exlicitly that the ole of the roagator is gauge indeendent at least u to one loo. The one loo fermion self-energy in Fig. 1 can be calculated in a general covariant gauge in a straightforward manner and in dimensional regularization it has a closed form exression in terms of hyergeometric functions. However, to see the singularity structure, we note that near the mass shell m it behaves like Σ 1) ) = 1 ) e D D m 3 D) D ) D 3 ) md 1 + ξ) 1 D m 1 ) D 3 ) + ξ/ 1 D m + + D + 1)ξ ) / D 1 + ) ξ)m ) D ) 1 D m ξ/ D 1 + ) ξ)m D ) ]. 4) 4 D) This shows that for D > 3, there are the usual ultraviolet divergences coming from 3 D) and D )), but there is no mass shell singularity so that the self-energy can have a roer Taylor exansion as in 3). On the other hand, when D 3there are not only infrared divergences the gamma function singularities in these cases can be identified with infrared divergences) but also mass shell singularities which show that a Taylor exansion of the self-energy is not ossible. Exlicitly, for examle, we see from 4) that for D = + ɛ the leading divergence of the self-energy has the form Σ 1) ) = e 1 + ξ) 1 4m m ) 1 1 ɛ. 5) This is infrared divergent, has a mass shell singularity and is also exlicitly gauge deendent. Although the leading divergence in 5) can be set to zero with the choice ξ = 1, subleading divergences which corresond to mass shell singularities) ersist. In fact, there is no value of ξ which will make the self-energy comletely well behaved. It is rather counterintuitive that the massive Schwinger model has an infrared divergence while the massless Schwinger model does not. This has its origin in the fact that the Dirac algebra in the self-energy in Fig. 1 leads to the numerator say, in the Feynman gauge for illustration uroses) γ μ / + m)γ μ = ɛ/ + + ɛ)m). Whenm 0, this numerator is finite in the limit ɛ 0 and does not moderate the infrared divergence arising from the integral while when m = 0, the numerator behaves as ɛ/ and the ɛ in the numerator moderates the infrared behavior making it finite. Since the one loo self-energy is divergent, it is worth asking if the higher loo corrections can cure the roblem. A general analysis of this question is rather difficult. Instead we will show that when summed to all loos in the quenched rainbow aroximation, the leading divergence in the fermion self-energy does indeed vanish. First let us recall that the overlaing divergence grahs, in general, have a softer infrared behavior 16] so that in studying the leading divergence structure these can be neglected. Second, let us note here that when infrared divergences are resent, normally including the olarization tensor in the hoton roagator at higher loos imroves the behavior. However, in the case of the massive Schwinger model, the one loo olarization tensor has the exlicit form Π μν = η μν μ ν ) Π ) T, 6) where Π T ) = e ] m dx m x1 x). 7) It is clear from 7) that when the fermion mass vanishes Π T ) = e giving a mass to the hoton in the massless) Schwinger model. When m 0, however, the function Π T ) is non-analytic. In fact, for 0 4m we obtain from 7) Π ) T = e 1 4m 4m 1 arctan ] 1, 8) 4m 1 while for 4m we have from 7) Π ) m T = e 1 1 4m ] 1 ln. 9) 1 4m m Eq. 9) allows us to take the massless limit yielding the result for the Schwinger model. On the other hand, for a nonzero fermion mass, it is 8) which allows us to take the limit 0leadingto Π ) T 0 e 1 1]=0, 10) showing that a hoton mass is not generated. The absence of a Higgs henomena in the massive Schwinger model is already known from a different analysis 1,13].) As a result, including vacuum olarization in the hoton roagator in higher orders does not hel and we restrict our analysis to the quenched rainbow aroximation see Fig. ).

3 416 A.K. Das et al. / Physics etters B ) and satisfies n μ D μν ) = 0 = D μν n ν. 15) Fig.. Fermion self-energy diagrams in the quenched rainbow aroximation. The leading divergence terms in the self-energy diagrams in Fig. for the massive Schwinger model can be calculated in a straightforward manner and the contribution at the n + 1)th loo turns out to be recursively related to the contribution at the nth loo in a simle manner Σ n+1) ) = Σ n) )Σ 1) 1 ) / m. 11) / m This simle recursive behavior of the leading divergences in the quenched rainbow aroximation allows us to sum the self-energy to all orders in this aroximation) and yields ) n Σ = Σ n+1) = Σ 1) Σ 1) 1 / m n=0 n=0 / m = Σ 1) 1 / m Σ 1) / m) 0. 1) / m This shows exlicitly that the leading divergence vanishes in the self-energy when summed to all orders in the quenched rainbow aroximation for any value of ξ ). This simle relation can also be seen in another way. We note that the leading term in the selfenergy to all orders can be written as an integral equation ga equation) of the form Σ ) = Σ 1) ) + e d D k ) γ μ 1 D / + /k m Σ + k) 1 / + /k γ ν D μν k). 13) m The leading infrared divergent contribution comes from the region k 0 and as / m this equation is solved in this regime by Σ ) = / m) 0. We have exlicitly seen the same behavior in scalar QED in dimensions in covariant gauges as well.) Of course, this does not say anything about the subleading divergences which, in rincile, are much harder to calculate since they involve the overlaing grahs. However, the vanishing of the leading divergence when summed) suggests that the fermion self-energy may be better behaved when calculated in a hysical gauge and this is what we discuss next. This can be understood as follows. In the covariant gauge, the fermion self-energy receives contribution from the unhysical as well as the hysical degrees of the hoton and a resummation may be necessary to see the cancellation between the contributions from these degrees of freedom. However, having seen the cancellation in the covariant gauge, it is clear that since a hysical gauge has only hysical degrees of freedom, there is no room for cancellation any more and the result should already be well behaved at one loo. In fact hysical gauges, such as the axial gauge, are known to tame the infrared divergence much better and if there is no infrared divergence at one loo as we will see shortly), summing the rainbow grahs would no longer be a meaningful aroximation. et us consider the massive Schwinger model in the homogeneous axial gauge n A = 0 where n μ denotes a fixed direction in sace time. In this case, the hoton roagator has the form id μν ) = i η μν n ) μ ν + n ν μ + n μ ν, n n ) 14) The oles of the roagator in 14) at n = 0 are conventionally handled through the PV rincial value) rescrition 17,18] which is known to soften the infrared behavior. It is also known that the PV rescrition may be roblematic 19] for time-like n > 0) and light-like n = 0) axial gauges. For this reason we restrict ourselves to the class of sace-like n < 0) axial gauges. It is worth noting here that in D =, the sace-like axial gauges also include the Coulomb gauge. This can be seen in two different but equivalent) ways. First, we note that if we choose the gauge A 1 = 0 or n μ = 0, 1), the only nontrivial comonent of the roagator in 14) is given by id 00 ) = i 1, 16) which coincides with the roagator in the Coulomb gauge. Alternatively, we note that even after we imose the Coulomb gauge 1 A 1 = 0 which imlies that A 1 = A 1 t), thereisaresidualgauge symmetry reserving this gauge choice, namely, A 1 A 1 = A 1t) + 1 Λt, x), 1 A 1 = 0. 17) This determines Λt, x) = xαt) + βt) where αt), βt) are arbitrary functions of t. This leads to see 17)) A 1 = A 1t) + αt), 18) and shows that if we choose αt) = A 1 t), A 1 = 0 which corresonds to the axial gauge. Coulomb gauge is known to be notoriously singular in D =. However, in considering this as belonging to the class of axial gauges in D =, it inherits the PV rescrition which gives it an imroved behavior. In D = thetwovectors n μ,n μ T = ɛμν n ν 19) define a comlete basis and, therefore, any vector including the Dirac matrices γ μ ) can be exanded in this basis. Using this the calculation of the fermion self-energy at one loo see Fig. 1) can be carried out in a manifestly covariant manner. The PV rescrition indeed makes the integral infrared finite and we obtain Σ 1),m,n) = e m + e ) / m)n γ + n γ / m) 4mn ) 1 + Ω ) 1 1 Ω ln 1 + Ω )], 0) 1 Ω where we have defined n Ω = n ). 1) m n The seemingly divergent second term in 0), in fact, vanishes as n 0 so that the self-energy is comletely well behaved. We also note that the comlete gauge deendence n μ deendence) of the self-energy is contained in the second term which vanishes as / m so that the one loo correction to the ole of the fermion roagator is gauge indeendent. This calculation can also be extended to two loos. The axial gauge calculations are, in general, extremely difficult at two loos and beyond. We have not yet been able to evaluate the two loo self-energy exlicitly. However, analyzing the integrand at this order including overlaing grahs) we find that the integral aears to be infrared finite. But, without

4 A.K. Das et al. / Physics etters B ) Fig. 3. The two three oint vertices F μ, G μ. an exlicit evaluation of the self-energy at two loos, it is not yet clear whether the ole of the roagator will be gauge indeendent at this order as well. As we have mentioned earlier, the gauge deendence of the ole of the roagator is best analyzed using Nielsen identities which can be derived using the BRST symmetry 0] of the theory. et us briefly sketch here how it is derived in a homogeneous axial gauge. The gauge fixed QED agrangian density including free ghosts) in any dimension is given by = 1 4 F μν F μν + ψ γ μ i μ ea μ ) m ) ψ + Fn A + cn c, ) where F denotes an auxiliary field introduced for gauge fixing. We can introduce the agrangian density of the standard sources needed to study BRST Ward) identities. However, in order to study the gauge deendence of the fermion two oint function we need to introduce an additional source H μ fermionic with ghost quantum number) for the comosite field ca μ ). Thus the comlete agrangian density for the sources is given by source = J μ A μ + JF + i ψχ χψ)+ i cη ηc) + ie Mcψ ψcm) + H μ ca μ, 3) where M, M denote resectively the sources bosonic sinors with anti-ghost/ghost quantum numbers) for the comosite BRST variations of the fields ψ and ψ. With these it is straightforward to derive the identity describing the gauge deendence of the fermion two oint function S 1 αβ ) n μ = F μ,αγ 0,, )Sγ 1 β ) + S 1 αγ )G μ,γ β0,, ), 4) where α,β,γ denote the Dirac indices and F μ, G μ denote the three oint vertices shown in Fig. 3. Using the form of the hoton roagator in 14), it is straightforward to check that D μν ) n λ = μ n D λν) ν n D μλ). 5) Furthermore, using this relation Eq. 5)) we can exlicitly check grahically u to two loos that the Nielsen identity 4) holds which is consistent with our earlier observation that there is no infrared divergence at least u to two loos. The Nielsen identity 4) can be studied order by order in a loo exansion. We note from 3) that there are no tree level three oint vertices corresonding to F μ,αγ, G μ,αγ see Fig. 3). They can arise only through radiative corrections and this is consistent with our exectation that gauge deendence in the fermion mass can ossibly arise at the loo level. The vertices F μ,αγ, G μ,αγ indeed arise beginning with one loo. Therefore, in the identity 4) at one loo, the fermion two oint functions on the right hand side have to be the ones at the tree level. As a result, the analysis simlifies and gauge indeendence of the fermion mass ole of the roagator) at one loo follows in a simle manner. This is, of course, comletely in agreement with our exlicit evaluation discussed in 0). At higher loos, however, the analysis is more involved than in covariant gauges. This is because in any non-covariant gauge involving a distinct tensor structure such as n μ ), the structure of S 1 ) becomes quite distinct from that of the tree level two oint function 1]. For examle, in two dimensions in the axial gauge we can exand the fermion two oint function at higher loos in a comlete basis of Dirac matrices A = 1, γ μ, γ 5 ), μ = 0, 1as S 1,m,n) = a A A = a I 1 + a μ γ μ + a P γ 5, 6) where the coefficients a A are, in general, orentz covariant functions of m, μ,n μ ). This has to be contrasted with the covariant gauges where the coefficients have to be functions of m, μ ) which makes the two oint function to be of the same form as in the tree level theory.) The deendence on a new structure makes it much harder to analyze the question of gauge indeendence of the ole. As a result, even though we know the Nielsen identity to hold at two loo level, we have not yet been able to conclude whether the ole of the roagator is gauge indeendent at that order. This is consistent with our earlier observation that even though there is no infrared divergence at two loos, we are unable to say if the ole is gauge indeendent without exlicitly evaluating the self-energy at two loos. A general study of the analysis of gauge indeendence directly from the Nielsen identity is resently underway. To summarize, we have shown exlicitly that the resence of infrared divergences as well as mass shell singularities can invalidate the conventional arguments for the gauge indeendence of the ole of the fermion roagator. We have shown this exlicitly in the case of the one loo fermion self-energy in the twodimensional) massive Schwinger model in covariant gauges. We have also shown that the leading divergence adds u to zero when all the diagrams are summed in the quenched rainbow aroximation. This suggests that the self-energy may be better behaved in a hysical gauge. We have shown that this is indeed the case in the sace-like) axial gauges where the PV rincial value) rescrition softens the infrared divergence and leads to a gauge indeendent ole of the roagator at one loo. We also find that the integrand in the self-energy) leads to an infrared finite integral at two loos although we have not yet been able to exlicitly evaluate the integral. Therefore, we are not yet able to say if the ole of the fermion roagator is gauge indeendent even at two loos. The same conclusions follow from an analysis of the Nielsen identity u to this order and this question is being further studied. If the axial gauge does allow a gauge indeendent definition of the dynamical fermion mass in the massive Schwinger model, it would indeed be worth trying such an analysis in the calculation of the quark mass in QCD. Details of our calculations will be ublished elsewhere.

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