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Effective action and tension renormalization for cosmic and fundamental strings
23 July 1998
Physics Letters B 432 Ž1998. 51–57
Effective action and tension renormalization for cosmic and fundamental strings Alessandra Buonanno a , Thibault Damour a
a,b
Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-YÕette, France b DARC, CNRS-ObserÕatoire de Paris, 92195 Meudon, France Received 5 March 1998; revised 1 May 1998 Editor: P.V. Landshoff
Abstract We derive the effective action for classical strings coupled to dilatonic, gravitational, and axionic fields. We show how to use this effective action for: Ži. renormalizing the string tension, Žii. linking ultraviolet divergences to the infrared Žlong-range. interaction between strings, Žiii. bringing additional light on the special cancellations that occur for fundamental strings, and Živ. pointing out the limitations of Dirac’s celebrated field-energy approach to renormalization. q 1998 Elsevier Science B.V. All rights reserved. PACS: 98.80.Cq
In many elementary particle models cosmic strings are expected to form abundantly at phase transitions in the early universe w1,2x. Oscillating loops of cosmic string might be a copious source of the various fields or quanta to which they are coupled. They might generate observationally significant stochastic backgrounds of: gravitational waves w3x, massless Goldstone bosons w4x, light axions w5,6x, or light dilatons w7x Žfor recent references on stochastic backgrounds generated by cosmic strings, see the reviews w1,2x.. An oscillating loop which emits outgoing gravitational, axionic or dilatonic waves, will also self-interact with the corresponding fields it has generated. This self-interaction is formally infinite if the string is modelled as being infinitely thin. Such infinite self-field situations are well known in the context of self-interacting particles. It was emphasized long ago by Dirac w8x, in the case of a classical
point-like electron moving in its own electromagnetic field, that the infinite self interaction problem is cured by renormalizing the mass: mŽ d . s mR y
e2
, Ž 1. 2d where mŽ d . is the Žultraviolet divergent. bare mass of the electron, m R the renormalized mass and d a cutoff radius around the electron. The analogous problem for self-interacting cosmic strings has been studied in Refs. w9–11x for the coupling to the axion field, in Ref. w12x for the coupling to the gravitational field, and in Ref. w13x for the coupling to the gravitational, dilatonic and axionic fields. See also Ref. w14x for the coupling to the electromagnetic field, in the case of superconducting strings. Related work by Dabholkar et al. w15,16x pointed out the remarkable cancellations, between the dilatonic, gravitational,
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 0 9 - 1
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
52
and axionic self-field effects, which take place for Žmacroscopic. fundamental strings. Though these cancellations can be derived for superstrings by appealing to supersymmetry Žand the existence of string-like BPS states w16x., they also take place for bosonic strings. It seems therefore useful to deepen their understanding without appealing to supersymmetry. The analog of the linearly-divergent renormalization Ž1. of the mass of a point particle is, for a string Žin four-dimensional spacetime., a logarithmicallydivergent renormalization of the string tension m , of the general form
m Ž d . s m R q C log
DR
ž / d
,
C s Cw q C g q CB .
Ž 2. The renormalization coefficient C is a sum of contributions due to each Žirreducible. field with which the string interacts. As above d denotes the ultraviolet cutoff length, while DR denotes an arbitrary renormalization length which must be introduced because of the logarithmic nature of the ultraviolet divergence. In this paper, we revisit the problem of the determination of the renormalization coefficient C Žwhich, as we shall see, has been heretofore uncorrectly treated in the literature. with special emphasis on: Ži. the streamlined extraction of C from the one-loop Žquantum and classical. effective action for self-interacting strings, namely Ž a and l denoting, respectively, the scalar and axionic coupling parameters; see Eq. Ž13. below. Cweffective - action s q4 a 2 G m2 ,
Ž 3a .
C geffective - action s 0 ,
Ž 3b .
CBeffective - action s y4 G l2 ,
Ž 3c .
Žii. the link between the ultraviolet divergence Ž2. and the infrared Žlong-range. interaction between strings, Žiii. the special cancellations that occur in C for fundamental Žsuper.-strings w15,16x, and Živ. the fact that the seemingly ‘‘clear’’ connection, pointed out by Dirac, between renormalization and field energy is valid only for electromagnetic and axionic fields but fails to give the correct sign and magnitude of C for gravitational and scalar fields.
In an independent paper, based on a quite different tensorial formalism w17,18x, Carter and Battye w19x, have reached conclusions consistent with ours for what concerns the vanishing of the gravitational contribution C g . wWe shall not consider here the finite ‘‘reactive’’ contributions to the equations of motion which remain after renormalization of the tension Žsee w10,11,20x..x The present work has been motivated by several puzzles concerning the various contributions to the renormalization coefficient C. First, Ref. w15x worked out the three contributions to the classical field energy around a straight Žinfinite. fundamental string and found a cancellation between two positiÕe and equal contributions due to w and B and a doubled negatiÕe contribution from gravity. We recall that Dirac emphasized that the cutoff dependence of the bare electron mass mŽ d . Žfor a fixed observable mass m R . was compatible with the idea that mŽ d . represents the total mass-energy of the particle plus that of the electromagnetic field contained within the radius d , so that mŽ d2 . y mŽ d1 . s q
d2 3
Hd
00 d x Tfield ,
Ž 4.
1
00 with Tfield s E 2rŽ8p . s e 2rŽ8p r 4 .. If we were to apply Dirac’s seemingly general result Ž4., the work of Ref. w15x Žgeneralized to arbitrary couplings a , l. would be translated into the following ‘‘field-energy’’ values of the renormalization coefficients: - energy 2 2 Cwfield expected s y4 a G m ,
Ž 5a .
- energy 2 C gfield expected s q8G m ,
Ž 5b .
- energy 2 CBfield expected s y4G l .
Ž 5c .
Only CBfield - energy agrees with CBeffective - action above. The sign of Cwfield - energy is wrong, as well as the value of C gfield - energy. Yet, the three partial C’s correctly cancel in the case of fundamental strings! ŽSee Eq. Ž15. below.. A second Žrelated. aspect of Eqs. Ž3a. – Ž3c. which needs to be understood concerns the vanishing of the gravitational contribution C geffective - action . Is this an accident or is there a simple understanding of it? A further puzzle is raised by the fact that the Žnonvanishing. value Ž5b. for C g was reproduced by the dynamical calculation of Ref. w13x.
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
To answer these puzzles we have computed the effective action obtained by eliminating to first order Žin a weak field expansion. the fields in the total action. To clarify the physical meaning of this effective action Žat both the quantum and classical levels. let us consider a generic action of the form Stot w z , A x s S0system w z x y 12 APy1A q JA ,
Ž 6.
where Py1 is the inverse of the propagator of the field A Žafter suitable gauge fixing., and where J w z x is the source of A Žwhich depends on the dynamical system described by the variables z .. We use here a compact notation which suppresses both integration over spacetime and any ŽLorentz or internal. labels on the fields: e.g. JA ' Hd n x J i Ž x . A i Ž x .. The quantum effective action for the dynamical system z arises when one considers processes where no real field quanta are emitted w21x. It is defined by integrating out the A field with trivial boundary conditions at infinity, namely expiS qeff w z x s ²0 Aout <0 Ain :z s D A exp Ž i S0 y 12 APy1A q JA
H
s expi S0 q 12 JPF J ,
.
H
R 2
p yie
,
Ž 8.
where du p s d n prŽ2p . n and R Žthe residue of the propagator. is a momentum-independent matrix R i j , when the field comes equipped with a ŽLorentz or internal. label: A i . The real part of the quantum effective action, Rew S qeff w z xx ' Sceff w z x, reads Sceff w z x s S0system w z x q S1 w z x , S1 w z x s 12 J w z x Psym J w z x , Psym ' Re w PF x s du p e i pŽ xyy. PP
H
with PP denoting the principal part. Sceff corresponds to a phase difference between the in-A-vacuum <0 Ain : and the out-A-vacuum <0 Aout :. On the other hand, twice the imaginary part of S qeff w z x gives the probability for the vacuum to remain vacuum: <²0 Aout <0 Ain :< 2 s expŽy2Im S qeff ., and is equal to the mean number of A-quanta emitted, n A s 2Im S qeff s p du pd Ž p 2 . J Ž yp . RJ Ž p . ,
H
Ž 11 .
where J Ž p . ' Hd n xeyi p x J Ž x . w23x. It is easily checked that Psym is nothing but the classical symmetric, half-retarded–half-advanced propagator. This shows that Rew S qeff x is the classical effective action, obtained by eliminating the field A in Ž6. by using the field equations written in the context of a classical non-dissipative system, i.e. a system interacting via half-retarded–half-advanced potentials. wIn the case of interacting point charges Sceff is the Fokker-ŽWheeler-Feynman. action.x Written more explicitly, the ‘‘one-classical-loop’’ Ži.e. one classical self-interaction. contribution S1 in Eq. Ž9. reads
Ž 7.
where the integration Žbeing Gaussian. is equivalent to estimating the integrand at the saddle-point, d Stotrd A 0 s yPy1A 0 q J s 0, and where, as is well known w21,22x, the trivial euclidean boundary conditions Žor the vacuum-to-vacuum prescription. translate into the appearance of the Feynman propagator. For massless fields in Feynman-like gauges, we can write PF Ž x , y . s du p e i pŽ xyy.
53
Ž 9. R
ž / p2
,
Ž 10 .
S1 w z x s 12
HHd
n
j x d n y J i Ž x . Pisym j Ž x, y. J Ž y.
s 12
HHd
n
x d n y Gsym Ž x , y . J i Ž x . R i j J j Ž y . ,
Ž 12 . Ž x, y . s where we used, from Eq. Ž10., Pisym j R i j Gsym Ž x, y ., Gsym being the symmetric scalar Green function: IGsym Ž x, y . s yd Ž n. Ž x y y .. It is easily checked Ž a posteriori . that varying with respect to the system variables z the classical effective action S0 w z x q S1w z x reproduces the correct equations of motion d Stot w z, A sym xrd z s 0 with A sym s Psym J being the classical half-retarded–half-advanced potential. Let us now apply this general formalism to string dynamics. We consider a closed Nambu-Goto string z m Ž s a . Žwith s a s Ž s 0 , s 1 .. interacting with gravitational gmn Ž x l . s hmn q hmn Ž x l ., dilatonic w Ž x . and axionic ŽKalb-Ramond. Bmn Ž x . fields. The action for this system is Stot s S s q S f , where a generic
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
54
action for the string coupled to gmn , w and Bmn reads S s s ym e 2 a w g d 2s y
H
'
l 2
HB
mn
dz m n dz n ,
Ž 13 . with g ' ydetga b Žga b ' gmn Ea z m E b z n ., and where the action for the fields is Sf s
1
n
16p G
Hd x'g Ž 14 .
with Hmn r s Em Bn r q En Br m q Er Bm n , g ' ydet Ž gmn . Žwe use the ‘‘mostly plus’’ signature.. Note that gmn is the ‘‘Einstein’’ metric Žwith a w-decoupled kinetic term g RŽ g .., while the ‘‘ string’’ metric Žor s-model metric. to which the s string is directly coupled is gmn ' e 2 a w gmn . The dimensionless quantity a parametrizes the strength of the coupling of the dilaton w to string matter, while the quantity l Žwith same dimension as the string tension m . parametrizes the coupling of Bmn to the string. The values of these parameters for fundamental Žsuper.-strings are, in n dimensional spacetime Žsee, e.g., w16x.,
'
(
lfs s m .
Ž 15 .
Unless otherwise specified we shall, for definiteness, work in n s 4 dimensions, so that a fs s 1. The additional coupling A ey4 a w in Eq. Ž14. between w and the kinetic term of the B-field is uniquely fixed by the requirement that w be a ‘‘dilaton’’ in the sense that a shift w ™ w q c be classically reabsorbable in a rescaling of the Žlength and mass. units, i.e. of gmn and the ŽEinstein-frame. gravitational constant G. In the present string case the spacetime sources J Ž x . of the previous generic formalism are worldsheet distributed i
J Ž x. s
d Sint d Ai Ž x .
S1 w z x s 12
As0
H (
s d 2s g 0 Ž z Ž s . . d Ž n. =Ž x y z Ž s . . J i Ž z . ,
Ž 16 .
2
HHd s
1
( (
d 2s 2 g 10 g 20
= Ž 4p Gsym Ž z 1 , z 2 . . CA Ž z 1 , z 2 . , CA Ž z 1 , z 2 . s
= R Ž g . y 2 = mw=m w y 121 ey4 a w Hmnr H mnr ,
a fs s 2r Ž n y 2 . ,
with g 0 s ydetga0b and ga0b ' hmn Ea z m E b z n . Inserting this representation into Eq. Ž12. leads Ž z 1m ' z m Ž s 1 . , z 2m ' z m Ž s 2 . ,g 10 ' g 0 Ž z 1 .. to
1 4p
Ž 17 .
J i Ž z1 . R i j J j Ž z 2 . .
Ž 18 .
The very general formula Ž17. will be our main tool for clarifying the paradoxes raised above. First, in 4 dimensional spacetime, the integral Ž17. diverges logarithmically as s 2a ™ s 1a. There are several ways to regularize this divergence. A simple, formal procedure, used in the previous literature w13,11x, is to use the explicit expression of the 4-dimensional symmetric Green function Gsym Ž z 1 , z 2 . s 1rŽ4p . d ŽŽ z 1 y z 2 . 2 . to perform the s 20 integration in Eq. Ž17., and then to regularize the s 21 integration by excluding the segment ydc - s 21 y s 11 - dc . Here, the conformal-coordinate-dependent quantity dc is linked 0 to the invariant cutoff d ' Žg 0 .1r4 dc s g 11 dc . Other procedures are to use the regularized Green reg Ž function Gsym z 1 , z 2 . s 1rŽ4p . d ŽŽ z 1 y z 2 . 2 q d 2 . w24,9x, or dimensional continuation w20x. We have checked that these different procedures lead to the same results. By comparing Ž17. to the zeroth-order string action S0 w z x s ymŽ d . Hd 2s 1 g 10 , it is easily seen that the coincidence-limit-divergent contributio n f r o m Ž 1 7 . g e n e r a te s th e te r m qlogŽ1rd . Hd 2s 1 g 10 CAŽ z 1 , z 1 . which renormalizes S0 w z x when CAŽ z, z . is independent of z, as it will be. In this case, we have the very simple link that the A-contribution to the renormalization coefficient C of Eq. Ž2. is simply equal to the coincidence limit of Eq. Ž18.:
(
(
(
CA s CA Ž z , z . s
1 4p
J i Ž z . Ri j J jŽ z . .
Ž 19 .
This result allows one to compute in a few lines the various C A ’s. The worldsheet-densities Jw Ž z ., Jgmn Ž z ., JBmn Ž z ., of the sources for w , gmn and Bmn Žlinearized around the trivial background Ž0,hmn ,0.. are easily obtained by varying Eq. Ž13. Že.g. Jw Ž x . s
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
d S s r dw Ž x . They read
ws 0
(
s Hd 2s g 0 Jw Ž z . d Ž x y z ...
Jw Ž z . s y2 a m s ya m gll ,
Ž 20a.
Jgmn Ž z . s y 12 m g mn ,
Ž 20b.
JBmn
Ž 20c.
1 2
Ž z. sy l e
mn
,
g mn ' g 0a b Ea z m E b z n ,
e mn ' e a b Ea z m E b z n , Ž 21 .
(
Ž e 10 s ye 01 s 1r g 0 . are the worldsheet metric and the Levi-Civita tensor, viewed from the external Žbackground. spacetime. The residue-matrices R i j are also simply obtained by writing the Žlinearized. field equations d Stotrd A s 0 in the form I A s yRJ. This yields R w Jw s 4p G Jw ,
Ž 22a.
g g Rmnrs Jgrs s 32p G Jmn y
ž
1 ny2
hmn Jlg l , Ž 22b .
/
B B Rmnrs JBrs s 32p G Jmn .
Ž 22c.
Applying Eq. Ž19. yields, in any dimension n main results 2
Cw s Ž G a 2 m2 . Ž y2 . s q4G a 2 m2 , C g s 2G m
gmn g
coordinates Ž s 0 , s 1 ., null worldsheet coordinates s "s s 0 " s 1. Indeed, in terms of such coordinates one finds the simple left-right factorized form Žtypical of closed-string amplitudes.
(g (g 0 1
0 2
fs Ctot Ž z1 , z 2 .
s 32G m 2 Ž Eq z 1m . Ž Eq z 2 m . Ž Ey z 1n . Ž Ey z 2 n . ,
where
2
55
mn
y
Ž gll .
2
ny2
s 4G m2
1
our
Ž 23a. ny4 ny2
,
Ž 23b. CB s 2G l2 emn e mn s y4G l2 .
Ž 23c.
In the four dimensional case this yields Eqs. Ž3a. – Ž3c.. Note that C g vanishes only in 4 dimensions. Note also that the sum Ctot s Cw q C g q CB vanishes for fundamental strings Žnon renormalization w15,16x., Eq. Ž15., in any dimension, but that for n / 4 it is crucial to include the non-vanishing gravitational contribution. The special nature of the coincidencelimit cancellations taking place for fundamental strings is clarified by using, instead of conformal
1 In n) 4 dimensions the leading ultraviolet divergences are AC d 4y n which poses the problem of studying also the subleading ones.
Ž 24 . where E " z m ' E z mrEs ". In the coincidence limit, z 1 s z 2 s z, the right-hand side of Eq. Ž24. vanishes because E " z m are null vectors Žthe Virasoro constraints reading Ž E " z m . 2 s 0.. Using our general result Ž17. we can now exhibit the link between the ultraÕiolet object C s C Ž z, z . and infrared, i.e. long-range, effects. Indeed, let us consider a system made of two straight and parallel Žinfinite. strings Žwith the same orientation of the axionic source e mn ., which are, at some initial time, at rest with respect to each other. The condition for this initial state of relative rest to persist is that the interaction energy between the two parallel strings be zero, or at least independent of their distance. But the interaction energy is just Žmodulo a factor y2 and the omission of a time integration. the effective action Ž17. in which z 1 runs on the first string, while z 2 runs on the second one. As, in the case of two straight and parallel strings, C Ž z 1 , z 2 . is independent of z 1 and z 2 , we see that the vanishing of the tension-renormalization coefficient C s C Ž z, z . Žinitially defined as an ultraviolet object. is equivalent, through the general formula Ž17., to the absence of long-range forces between two parallel strings Žwhich is an infrared phenomenon.. This result allows us not only to make the link with the infrared-based arguments of Refs. w15,16x and notably with the no-longrange force condition discussed in Ref. w16x Žwhere they find, in 4-dimensions, a compensation between attractive scalar forces and repulsive axial ones., but also to understand in simple terms why the gravitational contribution to C vanishes: this is simply related to the fact that, in 4 dimensions, straight strings exert no gravitational forces on external masses. Summarizing in symbols, we have shown that effective - action long - range - force Cultraviolet s Cinfrared . We have also independently verified, by a direct calculation of the
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
56
string equations of motion, that there were errors in the dynamical calculations of Ref. w13x and that the correct result was indeed given by Eqs. Ž23a. – Ž23c. dynamical effective - action w20x, so that, in symbols, Cultraviolet s Cultraviolet . As is discussed in detail in Ref. w20x, the main problem with the dynamical calculations of Ref. w13x Žbesides some computational errors for the dilaton force. is that the equations of motion for self-interacting strings, without external forces, are sufficient to prove renormalizability, but do not contain enough information for extracting the value of the tension renormalization. To determine unambiguously the renormalization of m one needs, either to explicitly couple the string to external Žsay, axionic. fields, or to work only with the strictly variational equations of motion d S srd z m. There remains, however, to understand the discrepancy between the dynamical C’s and the expected field-energy ones, Eqs. Ž5a. – Ž5c.. This puzzle is resolved by noting that the coupling of a string to Bmn Žas well as the coupling of a point particle to Am considered by Dirac. is the only one to be metric-independent, SBint s y 12 l HBmn dz m n dz n , and therefore the only one not to contribute to the total mn stress-energy tensor Ttot s 2 gy1 r2 d Srd gmn . By contrast, for the fields w and gmn the total interaction energy cannot be unambiguously localized only in the field, there are also interaction-energy contributions which are localized on the sources. These Ždivergent. source-localized interaction-energies are included in the effective action S1w z x but are missed mn in Tfield - energy , thereby explaining the discrepancies for Cwfield - energy and C gfield - energy. To conclude, let us summarize the new results of this work. We have derived the ‘‘one-classical-loop’’ Ži.e. one classical self-interaction. effective action for Nambu-Goto strings interacting via dilatonic, gravitational and axionic fields. Its explicit form, obtained by inserting Eqs. Ž20a. – Ž20c. and Eqs. Ž22a. – Ž22c. into Eq. Ž17. and Eq. Ž18., reads in any spacetime dimension n, Sceff w z x s ym Ž d . q 12
Hd s (g 2
1
2
HHd s
1
0 1
( (
d 2s 2 g 10 g 20
= Ž 4p Gsym Ž z 1 , z 2 . . Ctot Ž z 1 , z 2 . ,
Ž 25 .
where Gsym Ž z 1 , z 2 . is the symmetric scalar Green function and Ctot Ž z 1 , z 2 . s Cw q C g Ž z 1 , z 2 . q CB Ž z 1 , z 2 . ,
Ž 26 .
with Cw s 4G a 2 m2 ,
Ž 27a.
C g Ž z 1 , z 2 . s 2G m2 gmn Ž z 1 . g mn Ž z 2 . 1 y ny2
gmm Ž z 1 . gnn Ž z 2 . ,
CB Ž z 1 , z 2 . s 2G l2 emn Ž z 1 . e mn Ž z 2 . .
Ž 27b. Ž 27c.
Here g mn Ž z . and e mn Ž z . are the worldsheet metric and the Levi-Civita tensor, viewed from the external ŽMinkowski. spacetime, Eq. Ž21.. In the special case of fundamental strings, Eq. Ž15., the integrand of the first order contribution to the effective action simplifies to the left-right factorized form Ž24., when written in terms of null worldsheet coordinates. In 4 dimensions, the coincidence limit Ž z 1 ™ z 2 . generates logarithmic divergences in the first-order contribution to Sceff which can be absorbed in a renormalization of the bare string tension m Ž d .. The explicit value of this renormalization is given by Eq. Ž2. and Eqs. Ž3a. – Ž3c.. A simple understanding of the physical meaning of the various field-contributions to the renormalization of m has been reached: Ži. the values and signs of the various contributions are directly related to the worldsheet sources and the propagators of the various fields, Eq. Ž18.; Žii. the effective action approach allows one to relate the long-range interaction energy, and thereby the long-range force, between two straight and parallel strings to the coefficient C of the logarithmic divergence in the string tension. wIn particular, this explains in simple terms why the gravitational contribution to C vanishes Žin 4 dimensions.x; Žiii. the previously emphasized vanishing of the tension renormalization coefficient C in the case of fundamental strings w15,16x is clarified in two ways: Ža. by relating it Žfollowing Žii.. to the absence of long-range force between parallel fundamental strings wa fact interpretable in terms of supersymmetric ŽBPS. statesx, and Žb. by exhibiting the new, explicit, left-right factorized form Ž24., which clearly vanishes in the coincidence limit because of the Virasoro constraints wa fact valid for the bosonic
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
string, independently of any supersymmetry argumentx; Živ. finally, a puzzling discrepancy between the signs of the renormalization coefficients expected from Dirac’s field-energy approach to renormalization, Eq. Ž4. and Eqs. Ž5a. – Ž5c., and the Žcorrect. signs obtained by the effective action approach has been clarified by emphasizing the necessary existence of source-localized interaction energies for fields which are not p-forms.
w6x w7x w8x w9x w10x w11x w12x w13x w14x
Acknowledgements We thank Brandon Carter and Gabriele Veneziano for instructive exchanges of ideas.
References
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Physics Letters B 432 Ž1998. 51–57
Effective action and tension renormalization for cosmic and fundamental strings Alessandra Buonanno a , Thibault Damour a
a,b
Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-YÕette, France b DARC, CNRS-ObserÕatoire de Paris, 92195 Meudon, France Received 5 March 1998; revised 1 May 1998 Editor: P.V. Landshoff
Abstract We derive the effective action for classical strings coupled to dilatonic, gravitational, and axionic fields. We show how to use this effective action for: Ži. renormalizing the string tension, Žii. linking ultraviolet divergences to the infrared Žlong-range. interaction between strings, Žiii. bringing additional light on the special cancellations that occur for fundamental strings, and Živ. pointing out the limitations of Dirac’s celebrated field-energy approach to renormalization. q 1998 Elsevier Science B.V. All rights reserved. PACS: 98.80.Cq
In many elementary particle models cosmic strings are expected to form abundantly at phase transitions in the early universe w1,2x. Oscillating loops of cosmic string might be a copious source of the various fields or quanta to which they are coupled. They might generate observationally significant stochastic backgrounds of: gravitational waves w3x, massless Goldstone bosons w4x, light axions w5,6x, or light dilatons w7x Žfor recent references on stochastic backgrounds generated by cosmic strings, see the reviews w1,2x.. An oscillating loop which emits outgoing gravitational, axionic or dilatonic waves, will also self-interact with the corresponding fields it has generated. This self-interaction is formally infinite if the string is modelled as being infinitely thin. Such infinite self-field situations are well known in the context of self-interacting particles. It was emphasized long ago by Dirac w8x, in the case of a classical
point-like electron moving in its own electromagnetic field, that the infinite self interaction problem is cured by renormalizing the mass: mŽ d . s mR y
e2
, Ž 1. 2d where mŽ d . is the Žultraviolet divergent. bare mass of the electron, m R the renormalized mass and d a cutoff radius around the electron. The analogous problem for self-interacting cosmic strings has been studied in Refs. w9–11x for the coupling to the axion field, in Ref. w12x for the coupling to the gravitational field, and in Ref. w13x for the coupling to the gravitational, dilatonic and axionic fields. See also Ref. w14x for the coupling to the electromagnetic field, in the case of superconducting strings. Related work by Dabholkar et al. w15,16x pointed out the remarkable cancellations, between the dilatonic, gravitational,
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 0 9 - 1
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
52
and axionic self-field effects, which take place for Žmacroscopic. fundamental strings. Though these cancellations can be derived for superstrings by appealing to supersymmetry Žand the existence of string-like BPS states w16x., they also take place for bosonic strings. It seems therefore useful to deepen their understanding without appealing to supersymmetry. The analog of the linearly-divergent renormalization Ž1. of the mass of a point particle is, for a string Žin four-dimensional spacetime., a logarithmicallydivergent renormalization of the string tension m , of the general form
m Ž d . s m R q C log
DR
ž / d
,
C s Cw q C g q CB .
Ž 2. The renormalization coefficient C is a sum of contributions due to each Žirreducible. field with which the string interacts. As above d denotes the ultraviolet cutoff length, while DR denotes an arbitrary renormalization length which must be introduced because of the logarithmic nature of the ultraviolet divergence. In this paper, we revisit the problem of the determination of the renormalization coefficient C Žwhich, as we shall see, has been heretofore uncorrectly treated in the literature. with special emphasis on: Ži. the streamlined extraction of C from the one-loop Žquantum and classical. effective action for self-interacting strings, namely Ž a and l denoting, respectively, the scalar and axionic coupling parameters; see Eq. Ž13. below. Cweffective - action s q4 a 2 G m2 ,
Ž 3a .
C geffective - action s 0 ,
Ž 3b .
CBeffective - action s y4 G l2 ,
Ž 3c .
Žii. the link between the ultraviolet divergence Ž2. and the infrared Žlong-range. interaction between strings, Žiii. the special cancellations that occur in C for fundamental Žsuper.-strings w15,16x, and Živ. the fact that the seemingly ‘‘clear’’ connection, pointed out by Dirac, between renormalization and field energy is valid only for electromagnetic and axionic fields but fails to give the correct sign and magnitude of C for gravitational and scalar fields.
In an independent paper, based on a quite different tensorial formalism w17,18x, Carter and Battye w19x, have reached conclusions consistent with ours for what concerns the vanishing of the gravitational contribution C g . wWe shall not consider here the finite ‘‘reactive’’ contributions to the equations of motion which remain after renormalization of the tension Žsee w10,11,20x..x The present work has been motivated by several puzzles concerning the various contributions to the renormalization coefficient C. First, Ref. w15x worked out the three contributions to the classical field energy around a straight Žinfinite. fundamental string and found a cancellation between two positiÕe and equal contributions due to w and B and a doubled negatiÕe contribution from gravity. We recall that Dirac emphasized that the cutoff dependence of the bare electron mass mŽ d . Žfor a fixed observable mass m R . was compatible with the idea that mŽ d . represents the total mass-energy of the particle plus that of the electromagnetic field contained within the radius d , so that mŽ d2 . y mŽ d1 . s q
d2 3
Hd
00 d x Tfield ,
Ž 4.
1
00 with Tfield s E 2rŽ8p . s e 2rŽ8p r 4 .. If we were to apply Dirac’s seemingly general result Ž4., the work of Ref. w15x Žgeneralized to arbitrary couplings a , l. would be translated into the following ‘‘field-energy’’ values of the renormalization coefficients: - energy 2 2 Cwfield expected s y4 a G m ,
Ž 5a .
- energy 2 C gfield expected s q8G m ,
Ž 5b .
- energy 2 CBfield expected s y4G l .
Ž 5c .
Only CBfield - energy agrees with CBeffective - action above. The sign of Cwfield - energy is wrong, as well as the value of C gfield - energy. Yet, the three partial C’s correctly cancel in the case of fundamental strings! ŽSee Eq. Ž15. below.. A second Žrelated. aspect of Eqs. Ž3a. – Ž3c. which needs to be understood concerns the vanishing of the gravitational contribution C geffective - action . Is this an accident or is there a simple understanding of it? A further puzzle is raised by the fact that the Žnonvanishing. value Ž5b. for C g was reproduced by the dynamical calculation of Ref. w13x.
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
To answer these puzzles we have computed the effective action obtained by eliminating to first order Žin a weak field expansion. the fields in the total action. To clarify the physical meaning of this effective action Žat both the quantum and classical levels. let us consider a generic action of the form Stot w z , A x s S0system w z x y 12 APy1A q JA ,
Ž 6.
where Py1 is the inverse of the propagator of the field A Žafter suitable gauge fixing., and where J w z x is the source of A Žwhich depends on the dynamical system described by the variables z .. We use here a compact notation which suppresses both integration over spacetime and any ŽLorentz or internal. labels on the fields: e.g. JA ' Hd n x J i Ž x . A i Ž x .. The quantum effective action for the dynamical system z arises when one considers processes where no real field quanta are emitted w21x. It is defined by integrating out the A field with trivial boundary conditions at infinity, namely expiS qeff w z x s ²0 Aout <0 Ain :z s D A exp Ž i S0 y 12 APy1A q JA
H
s expi S0 q 12 JPF J ,
.
H
R 2
p yie
,
Ž 8.
where du p s d n prŽ2p . n and R Žthe residue of the propagator. is a momentum-independent matrix R i j , when the field comes equipped with a ŽLorentz or internal. label: A i . The real part of the quantum effective action, Rew S qeff w z xx ' Sceff w z x, reads Sceff w z x s S0system w z x q S1 w z x , S1 w z x s 12 J w z x Psym J w z x , Psym ' Re w PF x s du p e i pŽ xyy. PP
H
with PP denoting the principal part. Sceff corresponds to a phase difference between the in-A-vacuum <0 Ain : and the out-A-vacuum <0 Aout :. On the other hand, twice the imaginary part of S qeff w z x gives the probability for the vacuum to remain vacuum: <²0 Aout <0 Ain :< 2 s expŽy2Im S qeff ., and is equal to the mean number of A-quanta emitted, n A s 2Im S qeff s p du pd Ž p 2 . J Ž yp . RJ Ž p . ,
H
Ž 11 .
where J Ž p . ' Hd n xeyi p x J Ž x . w23x. It is easily checked that Psym is nothing but the classical symmetric, half-retarded–half-advanced propagator. This shows that Rew S qeff x is the classical effective action, obtained by eliminating the field A in Ž6. by using the field equations written in the context of a classical non-dissipative system, i.e. a system interacting via half-retarded–half-advanced potentials. wIn the case of interacting point charges Sceff is the Fokker-ŽWheeler-Feynman. action.x Written more explicitly, the ‘‘one-classical-loop’’ Ži.e. one classical self-interaction. contribution S1 in Eq. Ž9. reads
Ž 7.
where the integration Žbeing Gaussian. is equivalent to estimating the integrand at the saddle-point, d Stotrd A 0 s yPy1A 0 q J s 0, and where, as is well known w21,22x, the trivial euclidean boundary conditions Žor the vacuum-to-vacuum prescription. translate into the appearance of the Feynman propagator. For massless fields in Feynman-like gauges, we can write PF Ž x , y . s du p e i pŽ xyy.
53
Ž 9. R
ž / p2
,
Ž 10 .
S1 w z x s 12
HHd
n
j x d n y J i Ž x . Pisym j Ž x, y. J Ž y.
s 12
HHd
n
x d n y Gsym Ž x , y . J i Ž x . R i j J j Ž y . ,
Ž 12 . Ž x, y . s where we used, from Eq. Ž10., Pisym j R i j Gsym Ž x, y ., Gsym being the symmetric scalar Green function: IGsym Ž x, y . s yd Ž n. Ž x y y .. It is easily checked Ž a posteriori . that varying with respect to the system variables z the classical effective action S0 w z x q S1w z x reproduces the correct equations of motion d Stot w z, A sym xrd z s 0 with A sym s Psym J being the classical half-retarded–half-advanced potential. Let us now apply this general formalism to string dynamics. We consider a closed Nambu-Goto string z m Ž s a . Žwith s a s Ž s 0 , s 1 .. interacting with gravitational gmn Ž x l . s hmn q hmn Ž x l ., dilatonic w Ž x . and axionic ŽKalb-Ramond. Bmn Ž x . fields. The action for this system is Stot s S s q S f , where a generic
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
54
action for the string coupled to gmn , w and Bmn reads S s s ym e 2 a w g d 2s y
H
'
l 2
HB
mn
dz m n dz n ,
Ž 13 . with g ' ydetga b Žga b ' gmn Ea z m E b z n ., and where the action for the fields is Sf s
1
n
16p G
Hd x'g Ž 14 .
with Hmn r s Em Bn r q En Br m q Er Bm n , g ' ydet Ž gmn . Žwe use the ‘‘mostly plus’’ signature.. Note that gmn is the ‘‘Einstein’’ metric Žwith a w-decoupled kinetic term g RŽ g .., while the ‘‘ string’’ metric Žor s-model metric. to which the s string is directly coupled is gmn ' e 2 a w gmn . The dimensionless quantity a parametrizes the strength of the coupling of the dilaton w to string matter, while the quantity l Žwith same dimension as the string tension m . parametrizes the coupling of Bmn to the string. The values of these parameters for fundamental Žsuper.-strings are, in n dimensional spacetime Žsee, e.g., w16x.,
'
(
lfs s m .
Ž 15 .
Unless otherwise specified we shall, for definiteness, work in n s 4 dimensions, so that a fs s 1. The additional coupling A ey4 a w in Eq. Ž14. between w and the kinetic term of the B-field is uniquely fixed by the requirement that w be a ‘‘dilaton’’ in the sense that a shift w ™ w q c be classically reabsorbable in a rescaling of the Žlength and mass. units, i.e. of gmn and the ŽEinstein-frame. gravitational constant G. In the present string case the spacetime sources J Ž x . of the previous generic formalism are worldsheet distributed i
J Ž x. s
d Sint d Ai Ž x .
S1 w z x s 12
As0
H (
s d 2s g 0 Ž z Ž s . . d Ž n. =Ž x y z Ž s . . J i Ž z . ,
Ž 16 .
2
HHd s
1
( (
d 2s 2 g 10 g 20
= Ž 4p Gsym Ž z 1 , z 2 . . CA Ž z 1 , z 2 . , CA Ž z 1 , z 2 . s
= R Ž g . y 2 = mw=m w y 121 ey4 a w Hmnr H mnr ,
a fs s 2r Ž n y 2 . ,
with g 0 s ydetga0b and ga0b ' hmn Ea z m E b z n . Inserting this representation into Eq. Ž12. leads Ž z 1m ' z m Ž s 1 . , z 2m ' z m Ž s 2 . ,g 10 ' g 0 Ž z 1 .. to
1 4p
Ž 17 .
J i Ž z1 . R i j J j Ž z 2 . .
Ž 18 .
The very general formula Ž17. will be our main tool for clarifying the paradoxes raised above. First, in 4 dimensional spacetime, the integral Ž17. diverges logarithmically as s 2a ™ s 1a. There are several ways to regularize this divergence. A simple, formal procedure, used in the previous literature w13,11x, is to use the explicit expression of the 4-dimensional symmetric Green function Gsym Ž z 1 , z 2 . s 1rŽ4p . d ŽŽ z 1 y z 2 . 2 . to perform the s 20 integration in Eq. Ž17., and then to regularize the s 21 integration by excluding the segment ydc - s 21 y s 11 - dc . Here, the conformal-coordinate-dependent quantity dc is linked 0 to the invariant cutoff d ' Žg 0 .1r4 dc s g 11 dc . Other procedures are to use the regularized Green reg Ž function Gsym z 1 , z 2 . s 1rŽ4p . d ŽŽ z 1 y z 2 . 2 q d 2 . w24,9x, or dimensional continuation w20x. We have checked that these different procedures lead to the same results. By comparing Ž17. to the zeroth-order string action S0 w z x s ymŽ d . Hd 2s 1 g 10 , it is easily seen that the coincidence-limit-divergent contributio n f r o m Ž 1 7 . g e n e r a te s th e te r m qlogŽ1rd . Hd 2s 1 g 10 CAŽ z 1 , z 1 . which renormalizes S0 w z x when CAŽ z, z . is independent of z, as it will be. In this case, we have the very simple link that the A-contribution to the renormalization coefficient C of Eq. Ž2. is simply equal to the coincidence limit of Eq. Ž18.:
(
(
(
CA s CA Ž z , z . s
1 4p
J i Ž z . Ri j J jŽ z . .
Ž 19 .
This result allows one to compute in a few lines the various C A ’s. The worldsheet-densities Jw Ž z ., Jgmn Ž z ., JBmn Ž z ., of the sources for w , gmn and Bmn Žlinearized around the trivial background Ž0,hmn ,0.. are easily obtained by varying Eq. Ž13. Že.g. Jw Ž x . s
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
d S s r dw Ž x . They read
ws 0
(
s Hd 2s g 0 Jw Ž z . d Ž x y z ...
Jw Ž z . s y2 a m s ya m gll ,
Ž 20a.
Jgmn Ž z . s y 12 m g mn ,
Ž 20b.
JBmn
Ž 20c.
1 2
Ž z. sy l e
mn
,
g mn ' g 0a b Ea z m E b z n ,
e mn ' e a b Ea z m E b z n , Ž 21 .
(
Ž e 10 s ye 01 s 1r g 0 . are the worldsheet metric and the Levi-Civita tensor, viewed from the external Žbackground. spacetime. The residue-matrices R i j are also simply obtained by writing the Žlinearized. field equations d Stotrd A s 0 in the form I A s yRJ. This yields R w Jw s 4p G Jw ,
Ž 22a.
g g Rmnrs Jgrs s 32p G Jmn y
ž
1 ny2
hmn Jlg l , Ž 22b .
/
B B Rmnrs JBrs s 32p G Jmn .
Ž 22c.
Applying Eq. Ž19. yields, in any dimension n main results 2
Cw s Ž G a 2 m2 . Ž y2 . s q4G a 2 m2 , C g s 2G m
gmn g
coordinates Ž s 0 , s 1 ., null worldsheet coordinates s "s s 0 " s 1. Indeed, in terms of such coordinates one finds the simple left-right factorized form Žtypical of closed-string amplitudes.
(g (g 0 1
0 2
fs Ctot Ž z1 , z 2 .
s 32G m 2 Ž Eq z 1m . Ž Eq z 2 m . Ž Ey z 1n . Ž Ey z 2 n . ,
where
2
55
mn
y
Ž gll .
2
ny2
s 4G m2
1
our
Ž 23a. ny4 ny2
,
Ž 23b. CB s 2G l2 emn e mn s y4G l2 .
Ž 23c.
In the four dimensional case this yields Eqs. Ž3a. – Ž3c.. Note that C g vanishes only in 4 dimensions. Note also that the sum Ctot s Cw q C g q CB vanishes for fundamental strings Žnon renormalization w15,16x., Eq. Ž15., in any dimension, but that for n / 4 it is crucial to include the non-vanishing gravitational contribution. The special nature of the coincidencelimit cancellations taking place for fundamental strings is clarified by using, instead of conformal
1 In n) 4 dimensions the leading ultraviolet divergences are AC d 4y n which poses the problem of studying also the subleading ones.
Ž 24 . where E " z m ' E z mrEs ". In the coincidence limit, z 1 s z 2 s z, the right-hand side of Eq. Ž24. vanishes because E " z m are null vectors Žthe Virasoro constraints reading Ž E " z m . 2 s 0.. Using our general result Ž17. we can now exhibit the link between the ultraÕiolet object C s C Ž z, z . and infrared, i.e. long-range, effects. Indeed, let us consider a system made of two straight and parallel Žinfinite. strings Žwith the same orientation of the axionic source e mn ., which are, at some initial time, at rest with respect to each other. The condition for this initial state of relative rest to persist is that the interaction energy between the two parallel strings be zero, or at least independent of their distance. But the interaction energy is just Žmodulo a factor y2 and the omission of a time integration. the effective action Ž17. in which z 1 runs on the first string, while z 2 runs on the second one. As, in the case of two straight and parallel strings, C Ž z 1 , z 2 . is independent of z 1 and z 2 , we see that the vanishing of the tension-renormalization coefficient C s C Ž z, z . Žinitially defined as an ultraviolet object. is equivalent, through the general formula Ž17., to the absence of long-range forces between two parallel strings Žwhich is an infrared phenomenon.. This result allows us not only to make the link with the infrared-based arguments of Refs. w15,16x and notably with the no-longrange force condition discussed in Ref. w16x Žwhere they find, in 4-dimensions, a compensation between attractive scalar forces and repulsive axial ones., but also to understand in simple terms why the gravitational contribution to C vanishes: this is simply related to the fact that, in 4 dimensions, straight strings exert no gravitational forces on external masses. Summarizing in symbols, we have shown that effective - action long - range - force Cultraviolet s Cinfrared . We have also independently verified, by a direct calculation of the
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
56
string equations of motion, that there were errors in the dynamical calculations of Ref. w13x and that the correct result was indeed given by Eqs. Ž23a. – Ž23c. dynamical effective - action w20x, so that, in symbols, Cultraviolet s Cultraviolet . As is discussed in detail in Ref. w20x, the main problem with the dynamical calculations of Ref. w13x Žbesides some computational errors for the dilaton force. is that the equations of motion for self-interacting strings, without external forces, are sufficient to prove renormalizability, but do not contain enough information for extracting the value of the tension renormalization. To determine unambiguously the renormalization of m one needs, either to explicitly couple the string to external Žsay, axionic. fields, or to work only with the strictly variational equations of motion d S srd z m. There remains, however, to understand the discrepancy between the dynamical C’s and the expected field-energy ones, Eqs. Ž5a. – Ž5c.. This puzzle is resolved by noting that the coupling of a string to Bmn Žas well as the coupling of a point particle to Am considered by Dirac. is the only one to be metric-independent, SBint s y 12 l HBmn dz m n dz n , and therefore the only one not to contribute to the total mn stress-energy tensor Ttot s 2 gy1 r2 d Srd gmn . By contrast, for the fields w and gmn the total interaction energy cannot be unambiguously localized only in the field, there are also interaction-energy contributions which are localized on the sources. These Ždivergent. source-localized interaction-energies are included in the effective action S1w z x but are missed mn in Tfield - energy , thereby explaining the discrepancies for Cwfield - energy and C gfield - energy. To conclude, let us summarize the new results of this work. We have derived the ‘‘one-classical-loop’’ Ži.e. one classical self-interaction. effective action for Nambu-Goto strings interacting via dilatonic, gravitational and axionic fields. Its explicit form, obtained by inserting Eqs. Ž20a. – Ž20c. and Eqs. Ž22a. – Ž22c. into Eq. Ž17. and Eq. Ž18., reads in any spacetime dimension n, Sceff w z x s ym Ž d . q 12
Hd s (g 2
1
2
HHd s
1
0 1
( (
d 2s 2 g 10 g 20
= Ž 4p Gsym Ž z 1 , z 2 . . Ctot Ž z 1 , z 2 . ,
Ž 25 .
where Gsym Ž z 1 , z 2 . is the symmetric scalar Green function and Ctot Ž z 1 , z 2 . s Cw q C g Ž z 1 , z 2 . q CB Ž z 1 , z 2 . ,
Ž 26 .
with Cw s 4G a 2 m2 ,
Ž 27a.
C g Ž z 1 , z 2 . s 2G m2 gmn Ž z 1 . g mn Ž z 2 . 1 y ny2
gmm Ž z 1 . gnn Ž z 2 . ,
CB Ž z 1 , z 2 . s 2G l2 emn Ž z 1 . e mn Ž z 2 . .
Ž 27b. Ž 27c.
Here g mn Ž z . and e mn Ž z . are the worldsheet metric and the Levi-Civita tensor, viewed from the external ŽMinkowski. spacetime, Eq. Ž21.. In the special case of fundamental strings, Eq. Ž15., the integrand of the first order contribution to the effective action simplifies to the left-right factorized form Ž24., when written in terms of null worldsheet coordinates. In 4 dimensions, the coincidence limit Ž z 1 ™ z 2 . generates logarithmic divergences in the first-order contribution to Sceff which can be absorbed in a renormalization of the bare string tension m Ž d .. The explicit value of this renormalization is given by Eq. Ž2. and Eqs. Ž3a. – Ž3c.. A simple understanding of the physical meaning of the various field-contributions to the renormalization of m has been reached: Ži. the values and signs of the various contributions are directly related to the worldsheet sources and the propagators of the various fields, Eq. Ž18.; Žii. the effective action approach allows one to relate the long-range interaction energy, and thereby the long-range force, between two straight and parallel strings to the coefficient C of the logarithmic divergence in the string tension. wIn particular, this explains in simple terms why the gravitational contribution to C vanishes Žin 4 dimensions.x; Žiii. the previously emphasized vanishing of the tension renormalization coefficient C in the case of fundamental strings w15,16x is clarified in two ways: Ža. by relating it Žfollowing Žii.. to the absence of long-range force between parallel fundamental strings wa fact interpretable in terms of supersymmetric ŽBPS. statesx, and Žb. by exhibiting the new, explicit, left-right factorized form Ž24., which clearly vanishes in the coincidence limit because of the Virasoro constraints wa fact valid for the bosonic
A. Buonanno, T. Damourr Physics Letters B 432 (1998) 51–57
string, independently of any supersymmetry argumentx; Živ. finally, a puzzling discrepancy between the signs of the renormalization coefficients expected from Dirac’s field-energy approach to renormalization, Eq. Ž4. and Eqs. Ž5a. – Ž5c., and the Žcorrect. signs obtained by the effective action approach has been clarified by emphasizing the necessary existence of source-localized interaction energies for fields which are not p-forms.
w6x w7x w8x w9x w10x w11x w12x w13x w14x
Acknowledgements We thank Brandon Carter and Gabriele Veneziano for instructive exchanges of ideas.
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