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Rho mesons in the skyrme model: An alternative approach
Volume 185, number 3,4
PHYSICS LETTERSB
19 February 1987
R H O M E S O N S I N THE SKYRME MODEL: AN A L T E R N A T I V E A P P R O A C H Ulf-G MEISSNER 1 Instttute of Theorettcal Physics, Umverslty of Regensburg, D-8400 Regensburg, Fed Rep Germany Received 22 August 1986, Revised manuscript received 15 October 1986
I consider an alternative to a recently proposed model by Adkms coupling the rho meson to the Skyrme sohton Instead of a quartlc pmn term a sixth-order term representing the exchange of very mass~veomega mesons stabilizes the sohton All parameters in the lagrangiancan be fixed by mesomc data and the static baryon properties are therefore predicted parameter-free These predictions are a few percent better than the correspondingpredlctmns of the mass~veplon Skyrmemodel
Wltten [ 1 ] has argued that an i m p r o v e d description of baryons as chiral sohtons can be achieved by adding more mesons to the underlying effective theory a n d determining all parameters of the effective meson lagranglan from mesonic data (coupling constants, decay widths, etc ) I n this spirit, Adklns [ 2 ] has recently explored a model where he couples an independent, SU (2) -valued rho meson to the Skyrme sohton I n contrast to the previous model of Adklns and N a p p l [ 3 ] incorporating the co-meson to stabahze the sollton, the p discussed by Adklns does not exhibit this feature Therefore his lagranglan still contains a quartac plon term with a n arbitrary coefficient e Adklns uses the plon decay constant F~ a n d e as parameters to get the n u c l e o n a n d A masses This prescription gives again (cf ref [3]) too low values for F~ and gA, both are o f f b y approximately - 4 5 % The other static b a r y o n properties are within 30% of the data, an accuracy usually attributed to the massive plonlc Skyrme model Here, I propose an alternative formulation to Adklns' model As has been first shown by Iketanl [ 4 ], the i n t r o d u c t i o n of p-mesons as massive gauge particles leads to a Skyrme quartlc term (in the heavy mass h m l t ) , a n d furthermore the n u c l e o n - n u c l e o n Interaction calculations by Jackson et al [ 5 ] show that the quartlc term behaves indeed like a p-meson, Work supported in part by Deutsche Forschungsgememschaft under contract number We 655/9-i 0370-2693/87/$ 03 50 © Elsevier Science Publishers B V (North-Holland Physics Publishing D i v i s i o n )
e g in the tensor channel the divergent plonic contrib u t i o n is cancelled by a p-meson-hke c o n t r i b u t i o n leading to a finite a n d small tensor potential I n addition, the coefficient e i n front of the quartlc term can only be crudely estimated from n n - s c a t t e r l n g data [ 6 ] A physically more appealing scenario m the spirit of Wltten's argument [ 1 ] is therefore to use a sixthorder plonlc term to stablhze the sohton with its parameter given from the meson sector This has originally been proposed m the context of the Skyrme model by Jackson et al [ 7 ] The constant in front of the sixth-order term is closely connected to the to N N couphng constant which is experimentally well determined The lagranglan I will discuss is entirely determined by chlral symmetry a n d its parameters can be fixed completely from mesonic data Let us now t u r n to the plonlc part of the lagrangian I will consider It IS given by E1
The model presented here has one apparent weakness, namely It does not givethe proper d-waveng-scattermglengthsbecause of the omission of the Skyrmeterm [8] I do not consider this as a senous problem for two reasons (1) The slmphclty of the model anyway does not allow a correct description of all mesonic processes one can think of, and after all, it xsbuild to model baryomc properties, (2) it has been shown by Marshall et al [9] that the philosophy of tying the parameters of the effective theory to scattering lengths obviously does not lead to a satisfactorydescriptionof baryons at all Therefore,ffthese simple models should make any sense, one has to give up on some of the mesomc input data as ~t ~sdone here 399
Volume 185, number 3,4
PHYSICS LETTERS B
1 2 + =]zF~Tr[O~U 0/~ U]-¼e~Tr[BuBU]
+,~-2 ~2 [Tr(U)-2]
19 February 1987 i
F(u)
(1)
where the first t e r m is the well-known n o n - h n e a r amodel, and F~ is the plon decay constant ( = 186 MeV) U=exp(l'r.~/F~) is an S U ( 2 ) matrix descrlbang the lsovector plon field The stabahzlng term ~ T r [BuBU ] is nothing b u t the heavy mass h m l t o f the o - s t a b l h z a t l o n discussed by Adkans and N a p p l [ 3 ] B u IS the b a r y o n current
"\~,
',',L,
BU= _ (1/24n2) euv~P × T r [ U* O, U U* O~ U U* Oa U],
(2) o
a n d the constant e62 can be related to the o N N - c o u phng constant go via 2 2 e62 =go/rno,
(3)
The e m p m c a l value ofg~o is given by g~/4n = 10-12 [10] The third term in (1), a p l o n mass term, exphcltly breaks chlral symmetry, I will take m~ on its e x p e n m e n t a l value Thas lagranglan alone gives n s e to a stable sohton, but does not exhibit any Pmeson effects at all I therefore a d d the following Pmeson part [ 2 ] G=-~Tr[RL
+ aTr[ Ru~Ol' U t . U.O~Ut]
(4)
The p-field is a 2 × 2 four-vector
R u =P8 +I~'P u , and
R u is the
(5) usual
(abehan)
field strength
Ru*=OUR~-O"R u mp is the mass o f the rho meson, it will also be taken on its e m p m c a l value The whole lagrangian £~ + ~ (except for the p l o n mass t e r m ) is lnvarlant u n d e r chlral transformations U~LUR t, R--.LRUR *, as well as u n d e r P, C, a n d T The n u m b e r o f degrees o f f r e e d o m o f the p-field has to be reduced to insure unit lSOSpln, therefore the chlrally lnvarlant constraint Tr[R~- U] = 0
(6)
has to be i m p o s e d The last t e r m in (4) gives the pninteraction, the coupling constant a can be determ i n e d from the decay width o f the p into two plons, one finds a = 0 0444
400
10
15
To calculate the semlclasslcal properties o f the sohton o f this theory one makes the following timei n d e p e n d e n t ansatze
U=exp[ l'c.PF( r) ] , R°=0,
Ru~] +1~moTr[R 2 u+ R u]
05
Fig 1 The pton field F(u) for various values of the ¢0NN-couphng g,o The dashed hne gives the result for gZ/@r=6 45, the sohd one for g~/4zr=10 and the dashed-dotted one for g~/4zr= 12 u is a dlmensmnlessvariable, u=F~r(F~=186 MeV)
R'=lVaea'"~U~(r),
(7)
which are consistent with the constraint ( 6 ) a n d the nature o f the meson fields T r a n s f o r m i n g to d i m e n sionless units via u=F~r, F(u) a n d ~(u) can be o b t a i n e d from m m l m l z l n g the statm sohton mass The results o f this procedure are shown in figs 1 a n d 2, for various values o f the coupling constant g,o go,=9=(Nc/2)gp~ is the value o f go, one obtains from gauging the W e s s - Z u m l n o t e r m [11], g ~ / 4 ~ = 10 and 12 cover the range o f the d a t a One should note that F(u) a n d ~(u) are rather insensitive to the choice o f go, Before discussing the static b a r y o n properties shown in table 1, some remarks are m order To calculate the nucleon a n d A-masses, one has to admbatlcally rotate the soliton and evaluate the m o m e n t o f inertia (for details see ref [2 ]) In this model, the m o m e n t o f lnertm sphts into two parts, I = T, [F,, ~] + I 2 [ F , ~,, ¢2]
(8)
where ~ ( u ) a n d ~2(u) are 1/No-suppressed excitations o f the p-field These excltatlons contribute only
Volume 185, number 3,4
PHYSICS LETTERS B
0
05
10
15
~(u) I
~
'
'
I 02 ~
z z / / ~N
0~
~"~\,
'
-0,
\ \\
-06
\
\~/
-~ ....
,d /
- 08
Fig 2 The rho-meson ~(u) for v a n o u s values of go (see fig 1 )
on the percent level to the m o m e n t of inertia [ 12 ], and therefore have been neglected To check this assumption for the model presented here, I have calculated these functions ~l(u) and ~2(u) for g2/4zr = 10 and find that their contribution to the nucleon mass is less than 10 MeV The axial vector coupling constant gA lS Independent o f the p-excitations and is given by gA=--~[
i dfl ¼(fl2F' + ~ s a n 2F) 0
;
+
0
_e62 F t ( s i n 2 F +
du 4-7(
a----r-
F'sln2F) ;
+ i dfl 2c~[3~ s l n 3 F + 2fl~ ' sin F 0
-(a~' +~)acosF.F'll
(9)
19 February 1987
The lSOSCalar hedgehog radius rH can be directly computed from the baryon density Let us now discuss the results shown in table 1 The masses of the nucleon and A come out too high, but this is an expected result The inclusion o f soft-plon corrections [ 13 ] will certainly cure this problem The NA-spllttlng is a bit too small, but still reasonably close to the empirical value The axial vector coupling constant gA lS also a bit too small, but the result is clearly better than the one of Adklns who finds g A = 0 65 The other static properties not shown in table 1 agree roughly with the results o f Adklns Since one can argue [2] that in these kind ofchlral sollton models the plon decay constant should be below its experimental value, I have also calculated the sollton properties for F ~ = 1 5 0 MeV For one finds (as expected) much lower masses for the nucleon and the A, but also gA decreases It is worth noting that for this value o f F~, no stable sollton exists for g~ = 9, so the stablhty of the sohton in this model is closely tied to the mesonic parameters taken around their empirical values This again stresses the intimate connection of chlral sobton models and nuclear physics wisdom Let me finally remark that the model presented here should be only considered as a precursor of the more complete model discussed in ref [ 11 ], but it is much easier to handle and already exhibits some interesting features hke the aforementioned close relationship between the stability o f the sollton and the values of the mesonic parameters
g2/4n=lO
I would hke to thank Andreas Wlrzba and Wolfram Welse for clarifying discussions and support
Table 1 Static baryon properties in the model The m e s o m c input data are the paon decay constant f= and the o~NN couphng M H is the mass of
the classical sohton (the hedgehog), MN the mass of the nucleon The NA-mass sphttlng, the lSOSCalarradms rH and the axial vector couplingconstant gA are also given g~/4n = 6 45 is the value for the m-coupling constant one obtains from gauging the Wess-Zumlno term, 1 e go = (where gp~ = 6 0)
(Nc/2)gp,~,
g2~/4n
f~ (MeV)
MH (MeV)
MN (MeV)
Ma-- MN (MeV)
rH (fm)
gA
6 45 10 12
93 93 93
1219 1390 1468
1290 1442 1503
285 206 180
0 50 0 55 0 58
0 89 1 12 1 23
10
75
1012
1069
230
0 49
0 89
401
Volume 185, number 3,4
PHYSICS LETTERS B
References [ 1 ] E Wltten, m Sohtons and elementarypartlclephyslcs, eds A Chodos, E Hadljmlchael and C Tze (World Scientific, Singapore, 1984) pp 306-312 [2] G S Adkms, Phys Rev D 33 (1986) 193 [3] G S Adlonsand C R Nappl, Phys Lett B 137 (1984) 251 [4] K Iketanl, Kyushu-report No KYUSHU-84-HE-2 (1984), unpubhshed [5] A Jackson, A Jackson and V Pasquler, Nucl Phys A432 (1985) 567
402
19 February 1987
[6] J F Donoghue, E Golowlch and B R Holstein, Plays Rev Lett 53 (1984) 747 [ 7 ] A Jackson, A D Jackson, A S Goldhaber, G E Brown and L C Castlllejo, Phys Lett B 154 (1985) 101 [8] T N Pham and T N Truong, Phys Rev D 31 (1985) 3027 [9] M Marshall et al, Phys Rev Lett 56 (1986) 436 [10] W Greta, Nucl Phys B 131 (1977) 255, D O Rlska and B J VerWest, Phys Lett Rev B 48 (1974) 17 [ 11 ] U -G Melssner and I Zahed, Phys Lett 56 (1986) 1035 [12] U - G Melssner, Phys Lett B 166 (1986) 169 [13] I Zahed, A Wlrzba and U -G Melssner, Phys Rev D 33 (1986) 830
PHYSICS LETTERSB
19 February 1987
R H O M E S O N S I N THE SKYRME MODEL: AN A L T E R N A T I V E A P P R O A C H Ulf-G MEISSNER 1 Instttute of Theorettcal Physics, Umverslty of Regensburg, D-8400 Regensburg, Fed Rep Germany Received 22 August 1986, Revised manuscript received 15 October 1986
I consider an alternative to a recently proposed model by Adkms coupling the rho meson to the Skyrme sohton Instead of a quartlc pmn term a sixth-order term representing the exchange of very mass~veomega mesons stabilizes the sohton All parameters in the lagrangiancan be fixed by mesomc data and the static baryon properties are therefore predicted parameter-free These predictions are a few percent better than the correspondingpredlctmns of the mass~veplon Skyrmemodel
Wltten [ 1 ] has argued that an i m p r o v e d description of baryons as chiral sohtons can be achieved by adding more mesons to the underlying effective theory a n d determining all parameters of the effective meson lagranglan from mesonic data (coupling constants, decay widths, etc ) I n this spirit, Adklns [ 2 ] has recently explored a model where he couples an independent, SU (2) -valued rho meson to the Skyrme sohton I n contrast to the previous model of Adklns and N a p p l [ 3 ] incorporating the co-meson to stabahze the sollton, the p discussed by Adklns does not exhibit this feature Therefore his lagranglan still contains a quartac plon term with a n arbitrary coefficient e Adklns uses the plon decay constant F~ a n d e as parameters to get the n u c l e o n a n d A masses This prescription gives again (cf ref [3]) too low values for F~ and gA, both are o f f b y approximately - 4 5 % The other static b a r y o n properties are within 30% of the data, an accuracy usually attributed to the massive plonlc Skyrme model Here, I propose an alternative formulation to Adklns' model As has been first shown by Iketanl [ 4 ], the i n t r o d u c t i o n of p-mesons as massive gauge particles leads to a Skyrme quartlc term (in the heavy mass h m l t ) , a n d furthermore the n u c l e o n - n u c l e o n Interaction calculations by Jackson et al [ 5 ] show that the quartlc term behaves indeed like a p-meson, Work supported in part by Deutsche Forschungsgememschaft under contract number We 655/9-i 0370-2693/87/$ 03 50 © Elsevier Science Publishers B V (North-Holland Physics Publishing D i v i s i o n )
e g in the tensor channel the divergent plonic contrib u t i o n is cancelled by a p-meson-hke c o n t r i b u t i o n leading to a finite a n d small tensor potential I n addition, the coefficient e i n front of the quartlc term can only be crudely estimated from n n - s c a t t e r l n g data [ 6 ] A physically more appealing scenario m the spirit of Wltten's argument [ 1 ] is therefore to use a sixthorder plonlc term to stablhze the sohton with its parameter given from the meson sector This has originally been proposed m the context of the Skyrme model by Jackson et al [ 7 ] The constant in front of the sixth-order term is closely connected to the to N N couphng constant which is experimentally well determined The lagranglan I will discuss is entirely determined by chlral symmetry a n d its parameters can be fixed completely from mesonic data Let us now t u r n to the plonlc part of the lagrangian I will consider It IS given by E1
The model presented here has one apparent weakness, namely It does not givethe proper d-waveng-scattermglengthsbecause of the omission of the Skyrmeterm [8] I do not consider this as a senous problem for two reasons (1) The slmphclty of the model anyway does not allow a correct description of all mesonic processes one can think of, and after all, it xsbuild to model baryomc properties, (2) it has been shown by Marshall et al [9] that the philosophy of tying the parameters of the effective theory to scattering lengths obviously does not lead to a satisfactorydescriptionof baryons at all Therefore,ffthese simple models should make any sense, one has to give up on some of the mesomc input data as ~t ~sdone here 399
Volume 185, number 3,4
PHYSICS LETTERS B
1 2 + =]zF~Tr[O~U 0/~ U]-¼e~Tr[BuBU]
+,~-2 ~2 [Tr(U)-2]
19 February 1987 i
F(u)
(1)
where the first t e r m is the well-known n o n - h n e a r amodel, and F~ is the plon decay constant ( = 186 MeV) U=exp(l'r.~/F~) is an S U ( 2 ) matrix descrlbang the lsovector plon field The stabahzlng term ~ T r [BuBU ] is nothing b u t the heavy mass h m l t o f the o - s t a b l h z a t l o n discussed by Adkans and N a p p l [ 3 ] B u IS the b a r y o n current
"\~,
',',L,
BU= _ (1/24n2) euv~P × T r [ U* O, U U* O~ U U* Oa U],
(2) o
a n d the constant e62 can be related to the o N N - c o u phng constant go via 2 2 e62 =go/rno,
(3)
The e m p m c a l value ofg~o is given by g~/4n = 10-12 [10] The third term in (1), a p l o n mass term, exphcltly breaks chlral symmetry, I will take m~ on its e x p e n m e n t a l value Thas lagranglan alone gives n s e to a stable sohton, but does not exhibit any Pmeson effects at all I therefore a d d the following Pmeson part [ 2 ] G=-~Tr[RL
+ aTr[ Ru~Ol' U t . U.O~Ut]
(4)
The p-field is a 2 × 2 four-vector
R u =P8 +I~'P u , and
R u is the
(5) usual
(abehan)
field strength
Ru*=OUR~-O"R u mp is the mass o f the rho meson, it will also be taken on its e m p m c a l value The whole lagrangian £~ + ~ (except for the p l o n mass t e r m ) is lnvarlant u n d e r chlral transformations U~LUR t, R--.LRUR *, as well as u n d e r P, C, a n d T The n u m b e r o f degrees o f f r e e d o m o f the p-field has to be reduced to insure unit lSOSpln, therefore the chlrally lnvarlant constraint Tr[R~- U] = 0
(6)
has to be i m p o s e d The last t e r m in (4) gives the pninteraction, the coupling constant a can be determ i n e d from the decay width o f the p into two plons, one finds a = 0 0444
400
10
15
To calculate the semlclasslcal properties o f the sohton o f this theory one makes the following timei n d e p e n d e n t ansatze
U=exp[ l'c.PF( r) ] , R°=0,
Ru~] +1~moTr[R 2 u+ R u]
05
Fig 1 The pton field F(u) for various values of the ¢0NN-couphng g,o The dashed hne gives the result for gZ/@r=6 45, the sohd one for g~/4zr=10 and the dashed-dotted one for g~/4zr= 12 u is a dlmensmnlessvariable, u=F~r(F~=186 MeV)
R'=lVaea'"~U~(r),
(7)
which are consistent with the constraint ( 6 ) a n d the nature o f the meson fields T r a n s f o r m i n g to d i m e n sionless units via u=F~r, F(u) a n d ~(u) can be o b t a i n e d from m m l m l z l n g the statm sohton mass The results o f this procedure are shown in figs 1 a n d 2, for various values o f the coupling constant g,o go,=9=(Nc/2)gp~ is the value o f go, one obtains from gauging the W e s s - Z u m l n o t e r m [11], g ~ / 4 ~ = 10 and 12 cover the range o f the d a t a One should note that F(u) a n d ~(u) are rather insensitive to the choice o f go, Before discussing the static b a r y o n properties shown in table 1, some remarks are m order To calculate the nucleon a n d A-masses, one has to admbatlcally rotate the soliton and evaluate the m o m e n t o f inertia (for details see ref [2 ]) In this model, the m o m e n t o f lnertm sphts into two parts, I = T, [F,, ~] + I 2 [ F , ~,, ¢2]
(8)
where ~ ( u ) a n d ~2(u) are 1/No-suppressed excitations o f the p-field These excltatlons contribute only
Volume 185, number 3,4
PHYSICS LETTERS B
0
05
10
15
~(u) I
~
'
'
I 02 ~
z z / / ~N
0~
~"~\,
'
-0,
\ \\
-06
\
\~/
-~ ....
,d /
- 08
Fig 2 The rho-meson ~(u) for v a n o u s values of go (see fig 1 )
on the percent level to the m o m e n t of inertia [ 12 ], and therefore have been neglected To check this assumption for the model presented here, I have calculated these functions ~l(u) and ~2(u) for g2/4zr = 10 and find that their contribution to the nucleon mass is less than 10 MeV The axial vector coupling constant gA lS Independent o f the p-excitations and is given by gA=--~[
i dfl ¼(fl2F' + ~ s a n 2F) 0
;
+
0
_e62 F t ( s i n 2 F +
du 4-7(
a----r-
F'sln2F) ;
+ i dfl 2c~[3~ s l n 3 F + 2fl~ ' sin F 0
-(a~' +~)acosF.F'll
(9)
19 February 1987
The lSOSCalar hedgehog radius rH can be directly computed from the baryon density Let us now discuss the results shown in table 1 The masses of the nucleon and A come out too high, but this is an expected result The inclusion o f soft-plon corrections [ 13 ] will certainly cure this problem The NA-spllttlng is a bit too small, but still reasonably close to the empirical value The axial vector coupling constant gA lS also a bit too small, but the result is clearly better than the one of Adklns who finds g A = 0 65 The other static properties not shown in table 1 agree roughly with the results o f Adklns Since one can argue [2] that in these kind ofchlral sollton models the plon decay constant should be below its experimental value, I have also calculated the sollton properties for F ~ = 1 5 0 MeV For one finds (as expected) much lower masses for the nucleon and the A, but also gA decreases It is worth noting that for this value o f F~, no stable sollton exists for g~ = 9, so the stablhty of the sohton in this model is closely tied to the mesonic parameters taken around their empirical values This again stresses the intimate connection of chlral sobton models and nuclear physics wisdom Let me finally remark that the model presented here should be only considered as a precursor of the more complete model discussed in ref [ 11 ], but it is much easier to handle and already exhibits some interesting features hke the aforementioned close relationship between the stability o f the sollton and the values of the mesonic parameters
g2/4n=lO
I would hke to thank Andreas Wlrzba and Wolfram Welse for clarifying discussions and support
Table 1 Static baryon properties in the model The m e s o m c input data are the paon decay constant f= and the o~NN couphng M H is the mass of
the classical sohton (the hedgehog), MN the mass of the nucleon The NA-mass sphttlng, the lSOSCalarradms rH and the axial vector couplingconstant gA are also given g~/4n = 6 45 is the value for the m-coupling constant one obtains from gauging the Wess-Zumlno term, 1 e go = (where gp~ = 6 0)
(Nc/2)gp,~,
g2~/4n
f~ (MeV)
MH (MeV)
MN (MeV)
Ma-- MN (MeV)
rH (fm)
gA
6 45 10 12
93 93 93
1219 1390 1468
1290 1442 1503
285 206 180
0 50 0 55 0 58
0 89 1 12 1 23
10
75
1012
1069
230
0 49
0 89
401
Volume 185, number 3,4
PHYSICS LETTERS B
References [ 1 ] E Wltten, m Sohtons and elementarypartlclephyslcs, eds A Chodos, E Hadljmlchael and C Tze (World Scientific, Singapore, 1984) pp 306-312 [2] G S Adkms, Phys Rev D 33 (1986) 193 [3] G S Adlonsand C R Nappl, Phys Lett B 137 (1984) 251 [4] K Iketanl, Kyushu-report No KYUSHU-84-HE-2 (1984), unpubhshed [5] A Jackson, A Jackson and V Pasquler, Nucl Phys A432 (1985) 567
402
19 February 1987
[6] J F Donoghue, E Golowlch and B R Holstein, Plays Rev Lett 53 (1984) 747 [ 7 ] A Jackson, A D Jackson, A S Goldhaber, G E Brown and L C Castlllejo, Phys Lett B 154 (1985) 101 [8] T N Pham and T N Truong, Phys Rev D 31 (1985) 3027 [9] M Marshall et al, Phys Rev Lett 56 (1986) 436 [10] W Greta, Nucl Phys B 131 (1977) 255, D O Rlska and B J VerWest, Phys Lett Rev B 48 (1974) 17 [ 11 ] U -G Melssner and I Zahed, Phys Lett 56 (1986) 1035 [12] U - G Melssner, Phys Lett B 166 (1986) 169 [13] I Zahed, A Wlrzba and U -G Melssner, Phys Rev D 33 (1986) 830