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Isospin breaking mass differences from QCD
Volume 116B, number 6
PHYSICS LETTERS
28 October 1982
ISOSPIN BREAKING MASS DIFFERENCES FROM QCD P. PASCUAL and R. TARRACH Department o f Theoretical Physics, University o f Barcelona, Diagonal 647, Barcelona.28, Spain
Received 14 June 1982
We study the contributions to the mass differences p - n, Z+ - ~ -, -.o _ .~- proportional to the isospin breaking QCD parameters to -~ md[m u - 1 and 6 =- (dd)/(fiu) - 1 within the ITEP sum rule approach. By taking into account the electromagnetic contribution as estimated with the Cottingham formula very good agreement with experiment is found for to "~ 0.8 and 8 --~0.006, which are very reasonable values.
1. The study o f the baryon masses within the framework o f the ITEP sum rules [1 ] has been recently started by Ioffe [2] and Chung et al. [3]. We have done a careful analysis o f the 1/2 + baryon octet and have found good values for the N, Z, ,~ and A masses in terms o f the SU(3) breaking parameters ms, the strange quark mass, and 3' = (~s)/(fu) - 1, the normalized difference o f the strange rr,inus nonstrange quark vacuum condensate [4]. SU(2) was supposed to be a good symmetry as well as chiral SU(2) × SU(2) so that m u = m d = 0 and (dd) = (flu). Our results for the most standard value o f (fu)l/3 = - 2 5 0 MeV were ms(1.3 GeV 2) -----210 MeV and (~s)1/3 "" 0.8 (fu)l/3, in fair agreement with kaon PCAC and with other estimates of strange quark masses. They led to the following values o f the 1/2 + baryon masses
differences M n - Mp, M z - - ME÷ and M ~ - - M~o as functions of the isospin breaking QCD parameters co = md[m u -- 1 and 8 = (]d)[(fu) - 1, both o f which are far from free. Indeed w is known to be bound by 0.6 ~< co ~< 1.0 [5] and 8 must be very small because o f isospin symmetry. The best results we find are for = 0.8 and 8 = - 0 . 0 0 6 0 (M n - Mp)QC D = 2.26 MeV
(2.05 -+ 0.30 MeV),
(ME_-M]c+)QCD = 7.38 MeV (7.81 + 0.31 MeV), (M~- - M.-o )QCD = 5.51 MeV (5.5 -+ 0,7 MeV), or 6 = - 0 . 0 0 6 2 (Mn - Mp)QC D = 2.47 MeV,
M N = 0.966 GeV
(0.939 GeV),
(M E _ - M~÷)QCD = 7.45 MeV,
M~ = 1.187 GeV
(1.193 GeV),
(M~_ - M.~o)QCD = 5.27 MeV,
M . = 1.336 GeV
(1.318 GeV),
M A = 1.082 GeV
(1.116 GeV),
(1)
where the experimental values are given in parentheses. The results are such that the approach has to be considered as basically sound. This has encouraged us to go one step farther and study the isospin breaking mass differences within exactly the same approach. Specifically we obtain the nonelectromagnetic mass 0 031-9163[82]0000-0000/$02.75 © 1982 North-Holland
(2)
(3)
The values in parentheses are the experimental values obtained by Gasser and Leutwyler [5] in which the electromagnetic selfenergies are estimated with the Cottingham formula, the errors indicating the size o f the inelastic contributions. We find the values shown in (1) and (2) or (3) impressive. 2. Let us briefly report how (2) and (3) have been obtained. A more detailed account o f the approach and how the values (1) were obtained is given in ref. [4] 443
Volume 116B, number 6
PHYSICS LETTERS
28 October 1982
One first has to obtain the lowest dimensional composite gauge invariant operator which has all the quantum numbers of the 1/2 + baryon octet. For the particles we are interested in, which all have one repeated flavour, there are two independent ones, which for the proton are
as an expansion in l[y and at the same time gives more weight to the pole as compared to the continuum in eq. (9) versus eq. (7), because of the exponentials in the continuum integral. Our approach now proceeds as follows. From eq. (9) one can write the pole mass as
o dU(x) =
M =/~2(y)//~l(y ) +H(y) =M(v) + H(y),
[u (xle d (xll suT(x),
0~du(x) = eo~or[uT(x)C 7 5 d~o(x)] ur(x ).
(4)
For the proton one therefore has to take the general combination
Op(X) = o du( ) + tO(U( )
(5)
and similar combinations for the n, Z +, ~ - , E - and E0. The next step is to consider the two point Green function of O(x), F(O) - - i f d 4 x exp (ip-x)(0l
T(O(x)O(O))lO),
(6)
where 10) is the physical vacuum and write its Lehmann representation F(#) = #Fl(q2 ) + F2(q2), _
Fl(q2)
[Z[ 2
+1?
q2 _ M2 ~rao,l IZI2
F2(q2 ) -
d oImFl(O) - q2 o '
f do
M +1
q2 _ M 2
ImF2(o )
°o
(7)
q2 _ o
The third step is to perform a Borel transformation to the previous expressions [1]. With the definition
where H(y) is due to the continuum. Since there is no positivity condition for H(v) one can choose the combination of operators O 1 and 0 2 such that H(Yo) = 0,
M'(Y0) = 0,
(11)
so that M = M(Y0).
(12)
Of course since H(y) is unknown one obtainsy 0 from the second of eqs. (11) and uses t as a free parameter in eq. (12) to be fixed by optimization of the masses obtained from it. What remains is to compute F/(q 2) in order to obtain M(y). This was done perturbatively and beyond perturbation theory keeping the quark vacuum cordensates to lowest order in QCD. The quartic quark vacuum condensate was saturated with the vacuum intermediate state so that the whole computation only depends on ms, (flu) and (is). One then gets the octet masses shown in (1) for t = -1.10 from eq. (12). Notice that the only assumption is that the mixing angle t is the same for the whole octet, i.e., that it does not introduce SU(3) breaking. What we have done here is to further introduce isospin breaking through the parameters co and 6. The sum of the up ar.d down quark masses has been fixed from the pion PCAC relation (m u + md) [(flu) + (rid)] = --n2f2m2n"
h ( v ) =-
lma -
-
'
(8)
q2/n=y
one readily obtains 1
oo
YFI(Y)= IZI2 e-M2/y +~ f doe-a/Y ImFl(o), a0 oo
y~O2(y) =MIZ[2
e_M'2/y+ 1 f doe_O/y im F2(o), 7r
(9)
(70
The Borel transform makes the perturbative and nonperturbative computation ofyiOi0,) converge quicker 444
(10)
(13)
The computation is exactly the same as in ref. [4], but one has to keep up and down masses as well as distinguish (flu) and (]d). Again the only assumption is that t does not break isospin. Let us give the results for the proton. For the other particles it is enough to exchange some of the flavours and one obtains immediately the corresponding expressions. With the notation (27r)4Fl(q2) =Alq41n ( - q 2) +A2q21n(-q2 ) + A 3 In (_q2) + A4/q2
+As/q4 + ....
(14)
Volume 116B, number 6
PHYSICS LETTERS
(27r)4F2(q 2) = B 1 q4 In ( - q 2) + B2 q 2 In (---q 2) + B 3 In (_q2) + B4[q2 + ...,
(14 cont'd)
where only those terms which survive the Borel transform have been kept, we have found A 1 = i~8(5 + 2t + 5t2), 2 A 2 - - i ~1 [3mu(3 + 2t + 3t 2) + 6mu rod(1 -- t 2)
+ m~(5 + 2t + 5t2)],
= 0.8,
+ (dd)[3mu(1 - t 2) + ~md(5 + 2t + 5t2)] }, A 4 = 27r4(fu)[(fu)~(1 - t) 2 - 2(dd)(1 - t2)], A 5 = 2rr4(fu) {(flu)[ - ,~m2(3 - 2t + 3t 2) m u m a ( ~ - t 2) + ~ m2(1- t) 2]
+ ( a d ) [ - 2 m 2 ( 1 - t 2) _ ½mumd(1 + t) 2 + m2(1 -- t2)] }, B1 = 1 [ _ 3mu( 1 _ t 2) + ½md(1 _ t)2], B 2 = ~ 7r2 [3(fu)(1 - t 2) - ½(dd)(1 - t)2], B 3 = ½~r2 ((flu)I- 6m2(1 - t 2) - 3mumd(3 + 2t + 3t 2) -- 6mc](1 -- t2)] + ( d d ) [ - 3m2(1 + t) 2 - 3mumd(1 -- t 2) 1 2 -- ~ md(1
- - t ) 21
),
B 4 = 2~r4(fiu)((fiu) [mu(1 - t 2) + ~ md(5 + 2t + 5t2)]
+ (ad~ [mu(3 + 2t + 3t 2) + rod(1 -- t2)] ).
cases studied in ref. [4], according to the value o f ( • ) , (fu)l/3 = - 2 0 0 , - 2 5 0 or - 3 0 0 MeV. The first two are the upper and lower bounds of the generally accepted values [1,6], the third one was the one which led to the best baryon octet masses. Given this nonstrange quark condensate we have taken t, m s and (§s) as given in ref. [4] and only considered ~ and 6 as free parameters, although strongly constrained by 0.6 ~< co ~< 1.0 and 161 "~ 1. The best values are obtained for (flu) 1/3 = - 2 5 0 MeV,
A 3 = ½~r2 ((fiu)[~ mu(1 + t) 2 + 3rod(1 -- t2)]
-
28 October 1982
(15)
This is all what is needed for the computation of the contribution to the n - p, Z - - Z+ and ~ - - ~0 mass differences due to the QCD isospin breaking parameters eo and 5. 3. We have obtained numerical results for the three
6 = - 0 . 0 0 6 0 to - 0 . 0 0 6 2 ,
(16)
and they are shown in (2) and (3). We have also checked that the remaining isospin breaking combination ~+ + N - - 2N 0 is indeed of higher order in 6 and m u - m d and thus negligible as required by the fact that its electromagnetic selfenergy estimate already gives the whole experimental value [51. Let us comment here on the sign of 6. It has been proven by the ITEP group [7] that the slope of the quark condensate with respect to the mass is infinite at zero mass and such that the absolute value increases with the mass. It is however clear that isospin symmetry, i.e. the smallness of 6, requires an immediate flatten-off of the slope at the value of the up quark mass already. Then nothing can be said about the sign of 6, because once the slope has to be small nothing prevents it from having crossed already its zero value making the down quark condensate already smaller than the up quark condensate in absolute value. Not only the 1/2 + baryon octet masses, but also their isospin breaking mass differences are perfectly explained in terms of the QCD sum rule approach for the following values of the quark masses and condensates
md/m u = 1.8,
ms(1.3 GeV 2) "" 210 MeV,
(i]d) 1/3 "" 0.998 (flu) 1/3 , (flu) 1/3 "" - 2 5 0 MeV.
(§s) 1/3 "" 0.8 (flu) 1/3 , (17)
The quark mass values have turned out to be just the standard ones, as well as the nonstrange quark condensate. For the quark condensates isospin breaking turns out to be very small SU(3) breaking much larger; 445
Volume 116B, number 6
PHYSICS LETTERS
however not much is known from other sources on them. We would like to thank Eduardo de Rafael for let. ting himself be half-heartedly convinced that it was not too crazy to deal with mass differences.
References [1] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448.
446
28 October 1982
[2] B.L. Ioffe, Nucl. Phys. B188 (1981) 317. [3] Y. Chung, H.G. Dosch, M. Kremer and D. Schall, Phys. Lett. 102B (1981) 175; Nuel. Phys. B197 (1982) 55. [4] P. Pascual and R. Tarrach, Univ. of Barcelona preprint UBFT-FP-5-82, submitted to Nucl. Phys. B. [5] J. Gasser and H. Leutwyler, University of Bern preprint, BUTP-6/1982. [6] C. Becchi, S. Narison, E. de Rafael and F.J. Yndur~in, Z. Phys. C8 (1981) 338. [7] V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nuel. Phys. B191 (1981) 301.
PHYSICS LETTERS
28 October 1982
ISOSPIN BREAKING MASS DIFFERENCES FROM QCD P. PASCUAL and R. TARRACH Department o f Theoretical Physics, University o f Barcelona, Diagonal 647, Barcelona.28, Spain
Received 14 June 1982
We study the contributions to the mass differences p - n, Z+ - ~ -, -.o _ .~- proportional to the isospin breaking QCD parameters to -~ md[m u - 1 and 6 =- (dd)/(fiu) - 1 within the ITEP sum rule approach. By taking into account the electromagnetic contribution as estimated with the Cottingham formula very good agreement with experiment is found for to "~ 0.8 and 8 --~0.006, which are very reasonable values.
1. The study o f the baryon masses within the framework o f the ITEP sum rules [1 ] has been recently started by Ioffe [2] and Chung et al. [3]. We have done a careful analysis o f the 1/2 + baryon octet and have found good values for the N, Z, ,~ and A masses in terms o f the SU(3) breaking parameters ms, the strange quark mass, and 3' = (~s)/(fu) - 1, the normalized difference o f the strange rr,inus nonstrange quark vacuum condensate [4]. SU(2) was supposed to be a good symmetry as well as chiral SU(2) × SU(2) so that m u = m d = 0 and (dd) = (flu). Our results for the most standard value o f (fu)l/3 = - 2 5 0 MeV were ms(1.3 GeV 2) -----210 MeV and (~s)1/3 "" 0.8 (fu)l/3, in fair agreement with kaon PCAC and with other estimates of strange quark masses. They led to the following values o f the 1/2 + baryon masses
differences M n - Mp, M z - - ME÷ and M ~ - - M~o as functions of the isospin breaking QCD parameters co = md[m u -- 1 and 8 = (]d)[(fu) - 1, both o f which are far from free. Indeed w is known to be bound by 0.6 ~< co ~< 1.0 [5] and 8 must be very small because o f isospin symmetry. The best results we find are for = 0.8 and 8 = - 0 . 0 0 6 0 (M n - Mp)QC D = 2.26 MeV
(2.05 -+ 0.30 MeV),
(ME_-M]c+)QCD = 7.38 MeV (7.81 + 0.31 MeV), (M~- - M.-o )QCD = 5.51 MeV (5.5 -+ 0,7 MeV), or 6 = - 0 . 0 0 6 2 (Mn - Mp)QC D = 2.47 MeV,
M N = 0.966 GeV
(0.939 GeV),
(M E _ - M~÷)QCD = 7.45 MeV,
M~ = 1.187 GeV
(1.193 GeV),
(M~_ - M.~o)QCD = 5.27 MeV,
M . = 1.336 GeV
(1.318 GeV),
M A = 1.082 GeV
(1.116 GeV),
(1)
where the experimental values are given in parentheses. The results are such that the approach has to be considered as basically sound. This has encouraged us to go one step farther and study the isospin breaking mass differences within exactly the same approach. Specifically we obtain the nonelectromagnetic mass 0 031-9163[82]0000-0000/$02.75 © 1982 North-Holland
(2)
(3)
The values in parentheses are the experimental values obtained by Gasser and Leutwyler [5] in which the electromagnetic selfenergies are estimated with the Cottingham formula, the errors indicating the size o f the inelastic contributions. We find the values shown in (1) and (2) or (3) impressive. 2. Let us briefly report how (2) and (3) have been obtained. A more detailed account o f the approach and how the values (1) were obtained is given in ref. [4] 443
Volume 116B, number 6
PHYSICS LETTERS
28 October 1982
One first has to obtain the lowest dimensional composite gauge invariant operator which has all the quantum numbers of the 1/2 + baryon octet. For the particles we are interested in, which all have one repeated flavour, there are two independent ones, which for the proton are
as an expansion in l[y and at the same time gives more weight to the pole as compared to the continuum in eq. (9) versus eq. (7), because of the exponentials in the continuum integral. Our approach now proceeds as follows. From eq. (9) one can write the pole mass as
o dU(x) =
M =/~2(y)//~l(y ) +H(y) =M(v) + H(y),
[u (xle d (xll suT(x),
0~du(x) = eo~or[uT(x)C 7 5 d~o(x)] ur(x ).
(4)
For the proton one therefore has to take the general combination
Op(X) = o du( ) + tO(U( )
(5)
and similar combinations for the n, Z +, ~ - , E - and E0. The next step is to consider the two point Green function of O(x), F(O) - - i f d 4 x exp (ip-x)(0l
T(O(x)O(O))lO),
(6)
where 10) is the physical vacuum and write its Lehmann representation F(#) = #Fl(q2 ) + F2(q2), _
Fl(q2)
[Z[ 2
+1?
q2 _ M2 ~rao,l IZI2
F2(q2 ) -
d oImFl(O) - q2 o '
f do
M +1
q2 _ M 2
ImF2(o )
°o
(7)
q2 _ o
The third step is to perform a Borel transformation to the previous expressions [1]. With the definition
where H(y) is due to the continuum. Since there is no positivity condition for H(v) one can choose the combination of operators O 1 and 0 2 such that H(Yo) = 0,
M'(Y0) = 0,
(11)
so that M = M(Y0).
(12)
Of course since H(y) is unknown one obtainsy 0 from the second of eqs. (11) and uses t as a free parameter in eq. (12) to be fixed by optimization of the masses obtained from it. What remains is to compute F/(q 2) in order to obtain M(y). This was done perturbatively and beyond perturbation theory keeping the quark vacuum cordensates to lowest order in QCD. The quartic quark vacuum condensate was saturated with the vacuum intermediate state so that the whole computation only depends on ms, (flu) and (is). One then gets the octet masses shown in (1) for t = -1.10 from eq. (12). Notice that the only assumption is that the mixing angle t is the same for the whole octet, i.e., that it does not introduce SU(3) breaking. What we have done here is to further introduce isospin breaking through the parameters co and 6. The sum of the up ar.d down quark masses has been fixed from the pion PCAC relation (m u + md) [(flu) + (rid)] = --n2f2m2n"
h ( v ) =-
lma -
-
'
(8)
q2/n=y
one readily obtains 1
oo
YFI(Y)= IZI2 e-M2/y +~ f doe-a/Y ImFl(o), a0 oo
y~O2(y) =MIZ[2
e_M'2/y+ 1 f doe_O/y im F2(o), 7r
(9)
(70
The Borel transform makes the perturbative and nonperturbative computation ofyiOi0,) converge quicker 444
(10)
(13)
The computation is exactly the same as in ref. [4], but one has to keep up and down masses as well as distinguish (flu) and (]d). Again the only assumption is that t does not break isospin. Let us give the results for the proton. For the other particles it is enough to exchange some of the flavours and one obtains immediately the corresponding expressions. With the notation (27r)4Fl(q2) =Alq41n ( - q 2) +A2q21n(-q2 ) + A 3 In (_q2) + A4/q2
+As/q4 + ....
(14)
Volume 116B, number 6
PHYSICS LETTERS
(27r)4F2(q 2) = B 1 q4 In ( - q 2) + B2 q 2 In (---q 2) + B 3 In (_q2) + B4[q2 + ...,
(14 cont'd)
where only those terms which survive the Borel transform have been kept, we have found A 1 = i~8(5 + 2t + 5t2), 2 A 2 - - i ~1 [3mu(3 + 2t + 3t 2) + 6mu rod(1 -- t 2)
+ m~(5 + 2t + 5t2)],
= 0.8,
+ (dd)[3mu(1 - t 2) + ~md(5 + 2t + 5t2)] }, A 4 = 27r4(fu)[(fu)~(1 - t) 2 - 2(dd)(1 - t2)], A 5 = 2rr4(fu) {(flu)[ - ,~m2(3 - 2t + 3t 2) m u m a ( ~ - t 2) + ~ m2(1- t) 2]
+ ( a d ) [ - 2 m 2 ( 1 - t 2) _ ½mumd(1 + t) 2 + m2(1 -- t2)] }, B1 = 1 [ _ 3mu( 1 _ t 2) + ½md(1 _ t)2], B 2 = ~ 7r2 [3(fu)(1 - t 2) - ½(dd)(1 - t)2], B 3 = ½~r2 ((flu)I- 6m2(1 - t 2) - 3mumd(3 + 2t + 3t 2) -- 6mc](1 -- t2)] + ( d d ) [ - 3m2(1 + t) 2 - 3mumd(1 -- t 2) 1 2 -- ~ md(1
- - t ) 21
),
B 4 = 2~r4(fiu)((fiu) [mu(1 - t 2) + ~ md(5 + 2t + 5t2)]
+ (ad~ [mu(3 + 2t + 3t 2) + rod(1 -- t2)] ).
cases studied in ref. [4], according to the value o f ( • ) , (fu)l/3 = - 2 0 0 , - 2 5 0 or - 3 0 0 MeV. The first two are the upper and lower bounds of the generally accepted values [1,6], the third one was the one which led to the best baryon octet masses. Given this nonstrange quark condensate we have taken t, m s and (§s) as given in ref. [4] and only considered ~ and 6 as free parameters, although strongly constrained by 0.6 ~< co ~< 1.0 and 161 "~ 1. The best values are obtained for (flu) 1/3 = - 2 5 0 MeV,
A 3 = ½~r2 ((fiu)[~ mu(1 + t) 2 + 3rod(1 -- t2)]
-
28 October 1982
(15)
This is all what is needed for the computation of the contribution to the n - p, Z - - Z+ and ~ - - ~0 mass differences due to the QCD isospin breaking parameters eo and 5. 3. We have obtained numerical results for the three
6 = - 0 . 0 0 6 0 to - 0 . 0 0 6 2 ,
(16)
and they are shown in (2) and (3). We have also checked that the remaining isospin breaking combination ~+ + N - - 2N 0 is indeed of higher order in 6 and m u - m d and thus negligible as required by the fact that its electromagnetic selfenergy estimate already gives the whole experimental value [51. Let us comment here on the sign of 6. It has been proven by the ITEP group [7] that the slope of the quark condensate with respect to the mass is infinite at zero mass and such that the absolute value increases with the mass. It is however clear that isospin symmetry, i.e. the smallness of 6, requires an immediate flatten-off of the slope at the value of the up quark mass already. Then nothing can be said about the sign of 6, because once the slope has to be small nothing prevents it from having crossed already its zero value making the down quark condensate already smaller than the up quark condensate in absolute value. Not only the 1/2 + baryon octet masses, but also their isospin breaking mass differences are perfectly explained in terms of the QCD sum rule approach for the following values of the quark masses and condensates
md/m u = 1.8,
ms(1.3 GeV 2) "" 210 MeV,
(i]d) 1/3 "" 0.998 (flu) 1/3 , (flu) 1/3 "" - 2 5 0 MeV.
(§s) 1/3 "" 0.8 (flu) 1/3 , (17)
The quark mass values have turned out to be just the standard ones, as well as the nonstrange quark condensate. For the quark condensates isospin breaking turns out to be very small SU(3) breaking much larger; 445
Volume 116B, number 6
PHYSICS LETTERS
however not much is known from other sources on them. We would like to thank Eduardo de Rafael for let. ting himself be half-heartedly convinced that it was not too crazy to deal with mass differences.
References [1] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448.
446
28 October 1982
[2] B.L. Ioffe, Nucl. Phys. B188 (1981) 317. [3] Y. Chung, H.G. Dosch, M. Kremer and D. Schall, Phys. Lett. 102B (1981) 175; Nuel. Phys. B197 (1982) 55. [4] P. Pascual and R. Tarrach, Univ. of Barcelona preprint UBFT-FP-5-82, submitted to Nucl. Phys. B. [5] J. Gasser and H. Leutwyler, University of Bern preprint, BUTP-6/1982. [6] C. Becchi, S. Narison, E. de Rafael and F.J. Yndur~in, Z. Phys. C8 (1981) 338. [7] V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nuel. Phys. B191 (1981) 301.