Cottingham formula and the pion electromagnetic mass difference at finite temperature

21 October 1999

Physics Letters B 465 Ž1999. 241–248

Cottingham formula and the pion electromagnetic mass difference at finite temperature M. Ladisa a

a,b,c

, G. Nardulli

b,c

, S. Stramaglia

b,c

´ Centre de Physique Theorique, Centre National de la Recherche Scientifique, UMR 7644, Ecole Polytechnique, ´ 91128 Palaiseau Cedex, France b Dipartimento di Fisica, UniÕersita` di Bari, Italy c Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy Received 12 July 1999; accepted 7 September 1999 Editor: R. Gatto

Abstract We generalize the Cottingham formula at finite ŽT / 0. temperature by using the imaginary time formalism. The Cottingham formula gives the theoretical framework to compute the electromagnetic mass differences of the hadrons using a dispersion relation approach. It can be also used in other contexts, such as non leptonic weak decays, and its generalization to finite temperature might be useful in evaluating thermal effects in these processes. As an application we compute the pqy p 0 mass difference at T / 0; at small T we reproduce the behaviour found by other authors: d m2 ŽT . s d m2 Ž0. q O Ž a T 2 ., while for moderate T, near the deconfinement temperature, we observe deviations from this behaviour. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction The aim of this paper is to present the generalization of the Cottingham formula w1x to finite temperature Žfor a review of field theory at finite temperature and density see w2x.. The Cottingham formula allows the evaluation of the time-ordered product of two hadronic currents between hadronic states by a Dispersion Relation ŽDR. and its main application is in the evaluation of electromagnetic mass differences of hadrons. The use of the Cottingham formula in this context predates Quantum Chromo Dynamics ŽQCD. w3x; by the advent of QCD some modifications of the original formalism were adopted: for example, the problem of the convergence was settled by the use of an ultraviolet cut-off m2 related to the renormalization procedure w4x. Furthermore, the use of effective

field theories: chiral perturbation theory w5x and heavy quark effective theory w6x, has introduced well defined theoretical schemes to evaluate the relevant Feynman diagrams. Besides the use in the context of the electromagnetic mass differences of hadrons, the Cottingham formula has ben also applied in other fields, mainly related to non leptonic weak processes: kaon decays w7,8x, K 0 y K 0 mixing w9x, parity violations in nucleon interactions w10x, B-meson processes w11x; therefore its extension to finite temperature may be of some interest to analyze the role of the thermal effects in these physical situations. Among the possible applications we shall pick up in this paper the pqy p 0 mass difference at finite temperature, a subject that has received a continuous attention both in the past w12x and in more recent times w13x.

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 5 5 - 2

M. Ladisa et al.r Physics Letters B 465 (1999) 241–248

242

2. Cottingham formula at finite temperature

After the change of variables Ž5. one obtains the Cottingham formula w1x:

As already stressed in the introduction, the main application of the Cottingham formula is for the calculation of the electromagnetic mass splitting of mesons. The mass shift of the meson M due to the electromagnetic interaction can be obtained by computing:

d m2 s

ie 2

d4q

2

H Ž 2p .

gmn 4

2

q qie

T mn Ž q, p . ,

e2

d m2 s

16p

Ž 1.

ž

q 1y

T mn Ž q, p . s i d 4 x eyi q x ² M Ž p . < T Ž J m Ž x . J n Ž 0 . . < M Ž p . :

H

Ž 2. describes the Compton scattering of a virtual photon of four momentum q m off the meson M of momentum p m and J m is the electromagnetic current. The Compton amplitude can be decomposed in terms of gauge invariant tensors as follows: T mn Ž q, p . s D 1mn T1 Ž q 2 , n . q D 2mn T2 Ž q 2 , n . ,

Ž 3.

where

D 2mn s

1 m

2

ž

q mq n q2

pmy

n q

2

, qm

/ž

pn y

n q2

qn ,

/

n s pq .

Ž 4. 2

The Lorentz invariant structure functions T1Ž q , n . and T2 Ž q 2 , n . depend, in the meson rest frame, by < q < and q 0 s mn , where m is the meson mass. For fixed < q <, the singularities in the complex q 0 plane are placed just below the positive real axis and just above the negative real axis; therefore one can perform a Wick rotation to the imaginary axis q 0 s ik 0 , without encountering any singularity. After this transformation, the integration involves only spacelike momenta for the photon: q 0 ™ ik 0 ,

3

q 2 ™ yQ 2 s y Ž k 02 q < q <

2

..

Ž 5.

dQ 2 Q

2

Hyq''QQ

2

2

(

dk 0 Q 2 y k 02

= y3T1 Ž yQ 2 ,imk 0 .

where the hadronic tensor

D 1mn s ygmn q

q`

H0

k 02 Q2

/

T2 Ž yQ 2 ,imk 0 . .

Ž 6.

As discussed in w4x and w6x, the Q 2 integration 2 s m 2 . m is the Quantum range is cut-off at Qmax Chromo Dynamics ŽQCD. renormalization mass scale, at which both the strong coupling constant a s and the quark masses m q have to be specified and roughly represents the onset of the scaling behaviour of QCD. As is well known the renormalization procedure introduces counterterms which cancel the infinite contribution induced by virtual particles with momenta larger than m : therefore its net effect is analogous to a cut-off of the Q 2 integral at m2 . For approximate calculations there is a residue smooth dependence on m , which in a complete calculation is exactly canceled by the m-dependence of the renormalized quark masses and strong coupling constant. Typical values of m are in the range of 1–3 GeV, corresponding to the onset of the scaling behaviour of QCD and to a mass scale significantly larger than all the hadronic masses. We wish now to generalize this formula at finite temperature; we choose to work in the imaginary time formalism, which corresponds to substitute the integral over the energies with a discrete sum over the so called Matsubara frequencies v n s 2p nT : q`

q`

Hy` dq

0

f Ž q 0 . ™ 2p T

Ý f Ž q 0 . < q si v 0

n

.

Ž 7.

nsy`

This substitution corresponds to insert in Ž6. the factor q`

2p T

Ý nsy`

d Ž k 0 y vn . .

Ž 8.

M. Ladisa et al.r Physics Letters B 465 (1999) 241–248

We therefore obtain, from Ž6. and Ž8.:

a

d m2 s

4p

2

H0qm

2

dQ

2

qw a x

2p T

Q2

(Q y v 2

Ý

2 n

nsy w a x

= y3T1 Ž yQ 2 , im v n .

v n2

ž

q 1y

Q2

/

T2 Ž yQ 2 , im v n . ,

Ž 9.

where as

(Q

2

. Ž 10 . 2p T w ax represents the maximum integer contained in the real number a. Eq. Ž9. generalizes the Cottingham formula at T / 0 and is the starting point of the application to the pqy p 0 mass difference at finite temperature to be discussed in the following section.

243

In order to evaluate the DR in the case of the pion electromagnetic mass difference we consider the contribution of the Born term Žthe p meson itself. and the J P s 1y resonances v . Other contributions might be in principle sizeable, however it is well known, since the work by Harari w3x, that the DR are sufficiently well convergent at T s 0 and the first two polar terms in Ž11. represent by far the largest contribution to the electromagnetic mass difference. This stems from the fact that the pqy p 0 mass difference takes contribution from the Isospin s 2 mass term, and the time ordered product of the two electromagnetic currents in this case has no singularities for x ™ 0 Žor, equivalently, for Q 2 ™ `., differently from other hadronic mass differences, such as the proton-neutron mass difference, that present a 1rx 2 light-cone singularity. For the same reason the limit m ™ ` could also be taken in this case. To compute the different contributions to Ž11. we consider the following matrix elements Ž q s pX y p .: m

m < q ²pq Ž pX . < Jem p Ž p . : s F Ž q 2 . Ž p q pX . , Ž 16 .

3. The pion electromagnetic mass difference at finite temperature The invariant amplitudes T1 and T2 satisfy dispersion relations ŽDR. in the n variable. The DR for T2 is unsubtracted, while T1 requires one subtraction w3,5x: T1 Ž q 2 , n . s T1 Ž q 2 ,0 .

n2

q`

dn X

2

Im T1 Ž q 2 , n X .

, Ž 11 . nX 2 nX 2yn 2 1 q` X 2 Im T2 Ž q 2 , n X . T2 Ž q 2 , n . s dn . Ž 12 . p 0 nX 2yn 2 After the change of variables Ž5. these equations become: q

p

H0

H

T1 Ž yQ 2 ,imk 0 . q` 2

s T1 Ž yQ ,0 . y m

2

k 02

dn X

H0

n

X

2 2

W1 Ž yQ 2 , n X .

n

X

2

qm

2

k 02

,

m < ² v Ž pX , e . < Jem p 0 Ž p . : s ih Ž q 2 . e m l rsel) qr ps ,

Ž 17 . where el is the v polarization vector and F, h are electromagnetic form factors. They can be written as follows: 1 F Ž q2 . s , Ž 18 . 1 y q 2rm2V hŽ q 2 . s

h Ž 0. 1 y q 2rm2V

T2 Ž yQ 2 ,imk 0 . s Wi Ž q 2 , n . s

1

p

H0

dn X

2

Im Ti Ž q 2 , n .

W2 Ž yQ 2 , n X .

n

X

2

qm

2

k 02

F Ž q 2 ,T . s 1 q

Ž i s 1,2 . .

= ,

2T 2 3 fp2

mp2

ž / T2

1

mr2

1 y q 2rmr2

T2 y

g0

g rp p Ž T . g rg Ž T .

Ž 14 . Ž 15 .

Ž 19 .

m V is a mass parameter that we identify with the r mass. Indeed both form factors can be obtained by assuming r dominance. This hypothesis allows to extrapolate the behaviour at finite temperature. Indeed at T / 0, but small, one has w14x

Ž 13 . q`

,

4 fp2

g0

mp2

ž / T2

.

Ž 20 .

M. Ladisa et al.r Physics Letters B 465 (1999) 241–248

244

This expression takes into account the effective charge of a pion in thermal medium containing other pions at the equilibrium. Here fp s 93 MeV, g 0 Ž x . is given by w14x: g0 Ž x . s

1

y2

`

2p 2

H0 dy (x

2

ž(

q y 2 exp x 2 q y 2 y 1 .

,

/

Ž 21 . while g rp p ŽT . and g rg ŽT . are coupling constants for the rpp and r-photon vertex respectively, computed taking into account the effects of the thermal bath. They are obtained in the soft pion limit, which is sufficiently accurate for our purposes w14x. These effects Žand the related fp ŽT . dependence. have been computed by several authors and with different methods Žsee, for example w14,15x and references therein.. The results are: g rp p Ž T . s g rp p 1 y

ž

g rg Ž T . s g rg 1 y

5T 2

g 2 0

12 fp

T2 12 fp2

/

.

mp2

ž / T2

,

Ž 22 . Ž 23 .

The values of these constants at T s 0 are related by the formula g rp p g rg s1 , Ž 24 . mr2 a relation that implements the idea of r dominance of the electromagnetic form factor. It is worth stressing that, independently of the thermal effects induced by electromagnetism, the pion mass gets a thermal self energy also by the strong interactions with the dilute pion gas. In general we shall not consider these effects since they are identical for pq and p 0 and cancel in the difference. In one case the thermal self energy acts however as an infrared regulator Žsee below.. We quote therefore this effect w13x:

ž

mp2 q,0 Ž T . s mp2 q,0 1 y

T2 6 fp2

/

.

Ž 25 .

We also observe that, because of the interaction with the thermal bath, pions can couple directly to the photon without the intermediate virtual r state: this effect is at the origin of the last term in Ž20.. This term, which only exists at T / 0, gives a non-

vanishing contribution to the form factor for Q 2 ™ q`, which might be an artifact of the effective theory employed to compute it. However, as the Q 2 integration is cut-off at m2 and the results depends smoothly on m , there is no need to correct for this effect and we shall assume Ž20. as it stands. Let us now turn to the form factor relative to the v intermediate state. We put hŽ0. s 2.6 GeVy1 , which can be derived by v ™ pg decay rate. As it will be clear in the sequel, a more precise determination of hŽ q 2 . is not necessary because of the smaller role played by the v intermediate state in the calculation. For the same reason we omit to include thermal effects in the vertex vpg that, to our knowledge, have not yet been computed. Using the matrix elements and the coupling constants already introduced, we can calculate the electromagnetic pion mass splitting. To start with, we consider the T s 0 case. The contributions of the different terms to the DR are as follows; the subtraction term T1Ž q 2 ,0. is given by: T1 Ž q 2 ,0 . s y2 F 2 Ž q 2 . q

m2 q 2 h2 Ž q 2 .

nR

.

Ž 26 .

The two structure functions W1, 2 Ž q 2 , n . that appear in Ž13. and Ž14. are given by: W1 Ž q 2 , n . s qn R Ž m2 q 2 y n R2 . h2 Ž q 2 . d Ž n 2 y n R2 . , Ž 27 .

ž

W2 Ž q 2 , n . s y2 m2 q 2 F 2 Ž q 2 . d n 2 y

q4 4

/

q q 2 m2n R h2 Ž q 2 . d Ž n 2 y n R2 .

Ž 28 .

where

nR s

mr2 y m 2 y q 2 2

.

It is worth stressing that the Born term contributes only to W2 Ž q 2 , n ., its contribution to T1 only being through T1Ž q 2 ,0.. These expressions can be used to compute the electromagnetic pion mass difference at T s 0. The results are reported in Table 1, for m s 2 and m s 3 GeV and for m s `. It may be useful to stress that the numerical results are remarkably insensitive to variations of the cut-off. They show show that the

M. Ladisa et al.r Physics Letters B 465 (1999) 241–248

where m 2 ŽT . is given in Ž25.. By inserting these results in Eq. Ž9. we can compute numerically the sums over the discrete energy. The numerical results will be discussed in the next section.

Table 1 Contributions to d m in MeV; d m exp s 4.6 MeV Contribution

m s 2 GeV

m s 3 GeV

m s`

p v total

3.56 0.09 3.65

3.82 0.11 3.93

4.05 0.13 4.18

pion contribution represents by far the dominant part of the mass difference. The small difference Žaround 10%. between the data and the theoretical result should be attributed to a few other poles in the dispersion relation, most notably the a1 resonance. In the chiral limit mp s 0 we get, for m s `: 2

dm s

3 a mr2 fr2 4p

ln 2

fp

ž

fr2 fr2 y fp2

/

Ž 30 .

that is obtained using the chiral limit and the Weinberg sum rules. The extra factor in Ž30. as compared to Ž29. is given by fr2

ln 2

fp

ž

fr2 fr2 y fp2

4. Electromagnetic mass difference in the chiral limit In the chiral limit the expression for the electromagnetic pion mass difference looks remarkably simple:

d m2 Ž T . s

3 a mr2

, Ž 29 . 4p which gives, numerically, d m s mpqy mp 0 s 3.8 MeV, to be compared to the the experimental result d m exp s 4.6 MeV. We can also compare it to the the result of Das et al. w16x:

d m2 s

/

6a

H0m

4p 2

=

Q 2 q mv2 y m 2

n R Ž n R2 q m2v n2 .

T™0

4p

Ž Q 2 y m2 q m2 Ž T . . Q 2n R

n R2 q m2v n2

,

H0m dQ F 1 a

2

nsy w a x

H0m dQ F 2

q1

q 4 m2v n2

Ž 32 .

2

H0m dQ F 2

2

(

1y

2

Hy1 dx'1 y x

3a

Ž yQ 2 .

Ý

2

2

2

qw a x

=

Q 2 y m2 q m2 Ž T . 2

Ž 34 .

2

2

a™`

4p

s 8 m F Ž yQ ,T . y m 2 h 2 Ž yQ 2 .

6a

= lim

,

T2 Ž yQ ,im v n .

Ž 33 .

where d m2 s d m2 ŽT s 0. and l is a coefficient. The absence of the term linear in T can be proved explicitly by performing an asymptotic expansion in 1rT of d m2 ŽT .. The term independent of T is given by

s

2

.

d m2 Ž T . s d m2 q lT 2 ,

4p

2

2

2 n

The behaviour of d m2 ŽT . at small T is as follows:

Ž 31 .

2

2

s

v n2 Ž n R2 q m2 Q 2 .

F 2 Ž yQ 2 ,T . 2p T

(Q y v

Ý

6a

s y2 F 2 Ž yQ 2 ,T . y m2 h 2 Ž yQ 2 . y

Q2

nsy w a x

T1 Ž yQ 2 ,im v n . 2 Q2

dQ 2

2

qw a x

lim d m2 Ž T . s

and is numerically equal to 1.24 for frrfp s 1.66. Let us now consider the analogous calculation at T / 0. We shall use in Ž9.

=

245

n2 a2

Ž yQ 2 . 2

Ž yQ 2 . ,

Ž 35 .

which coincides with d m 2 , i.e. with the result obtained by putting directly T s 0 from the very beginning. As to the coefficient of the term in T 2 , let us write it as follows:

l s L1 q L 2 ,

Ž 36 .

M. Ladisa et al.r Physics Letters B 465 (1999) 241–248

246

as it arises from two different sources. The first term, L 1 , is obtained by expanding F 2 ŽyQ 2 ,T . in T and taking into account that g 0 Ž0. s 1r12: F 2 Ž yQ 2 ,T . s F 2 Ž yQ 2 . T2 y

F 2 Ž yQ 2 . q

8 fp2

F Ž yQ 2 .

3a T

3

p

qO ŽT 2 . ;

T2

ž

8 fp2

mr2 q m2

1q

3 m2

ln

mr2 q m2 mr2

/

,

Ž 38 . and, therefore, T2 L1 s y

8 fp2

mr2 q m2

ž

2

2

dQ 2

H0m (Q

2

F 2 Ž yQ 2 . ,

d m Ž 0. 1 q

3 m2

ln

mr2 q m2 mr2

/

.

Ž 39 .

but, given the range of the possible values of m , such a behaviour cannot be reached, as the deconfinement process should take place at much smaller value of T. For T ) 100 MeV numerical deviations from the behaviour Ž43. are therefore expected, as we will discuss below. We can compare these results with those obtained by the authors of w13x in the framework of chiral perturbation theory at finite temperature within the hard thermal loop approximation:

d m2 Ž T . s d m2 Ž 0. 1 y

The second term, i.e. L 2 , is obtained by putting F 2 Ž yQ 2 ,T . s F 2 Ž yQ 2 .

Ž 40 .

and computing 6a

L 2 s lim

T™0

=

2

4p T qw a x

1 a

2

2

H0m dQ F 2

(

1y

Ý nsy w a x

2

Ž yQ 2 .

n2 a

2

p y 2

,

Ž 41 .

(

where, as before, a s Q 2 r2p T. We have been unable to evaluate analytically this limit; however, when calculated numerically, for m G 0.5–1 GeV, the limit is basically independent of m and within a few percent is given by: L 2 f pa .

Ž 42 .

Putting the two contributions together we get

d m2 Ž T . f d m2 Ž 0. = 1y

T2 6 fp2

q pa T 2 .

ž

3 q 4

Ž 44 .

Ž 37 .

from this equation one gets the following contribution to d m 2 ŽT .

d m2 Ž 0. 1 y

We observe however that the behaviour of the last term in Eq. Ž43. holds only at small T ŽT F 100 MeV.. For T G mr2p , its T-dependence would be linear:

mr2 q m2 4m 2

ln

mr2 q m2 mr2

/ Ž 43 .

T2 6 fp2

q pa T 2 .

Ž 45 .

A part from the difference in d m2 Ž0. discussed above, we get a small deviation also in the correction A T 2 : whereas the term qpa T 2 is identical, the remaining part A T 2 differs by the factor 3 q 4

mr2 q m2 4m 2

ln

mr2 q m2 mr2

,

Ž 46 .

whose numerical value is 1.3 at m s 2 GeV and 1.5 at m s 3 GeV. This difference arises from the different treatment of the virtual photon effect: in particular the logarithmic m dependence in Ž43. arises from the direct photon coupling to pions, which is allowed by the pion electromagnetic form factor at T / 0, as given by Ž20.. In particular if one assumes that also this direct coupling, i.e. the last term proportional to T 2 in Ž20., is multiplied by the factor Ž1 y q 2rmr2 .y1 , then the same result as in w13x is obtained. Let us now turn to the numerical results of our analysis. In Fig. 1 we report two curves. The solid line is the result of the full numerical analysis contained in Section 3, while the dashed line represents d m2 ŽT . as computed by the approximate formula

M. Ladisa et al.r Physics Letters B 465 (1999) 241–248

Fig. 1. The difference d m2 s mp2 qy mp2 0 as a function of the temperature. The solid line is the result of the full calculation Žwith a value of the ultraviolet cut-off m s 3 GeV., the dashed line gives the result in the chiral and small temperature limit.

247

tion of time ordered products of two currents between hadronic states; even though it is generally applied to the calculation of electromagnetic mass differences of hadrons, its possible range of applications is wider and includes the evaluation of matrix elements of other products of currents such as those occurring in the analysis of weak non leptonic decays. Therefore its extension to T / 0 may permit the study of finite temperature effects also for weak hadronic processes. As an application, we have considered the finite temperature pqy p 0 electromagnetic mass difference; the use of the Cottingham formula leads to results similar to those reached by chiral perturbation theory for small temperatures 100 MeV. We have also computed deviations due to chiral symmetry breaking, which are of the order of 7% or less, and corrections to the hard thermal loop approximation whose role is relevant for T G 100 MeV.

Acknowledgements Ž43.. Both curves are obtained at m s 3 GeV. The comparison shows that: 1. For small T, the small deviation Ž- 7%. is mainly due to a chiral symmetry breaking term which is taken into account in the full calculation Žsolid line. and not in the approximate calculation Ždashed line.. This extra term is part of the pion pole contribution to the dispersion relation Žthe v contribution is always very tiny.. 2. For large T, i.e. T ) 100 MeV, the difference is mainly due to the fact that the approximation Ž43. gets worse with the increasing temperature; in particular the positive contribution, i.e. the last term in Ž43., is quadratic only for T F 100 MeV, while for T ) 100 MeV, it increases more slowly and the effect of the negative term remains unbalanced. For T s 200 MeV this correction is of the order of 80%. All these results are almost independent of m in the range 2–3 GeV.

5. Conclusions We have extended the Cottingham formula at finite temperature. This formula allows the computa-

We thank M. Pellicoro for useful discussions. One of us ŽM.L.. expresses his gratitude to T.N. Pham for his encouragements and hospitality.

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