Cosmic strings in string perturbation theory

  • Published on
    06-Jul-2016

  • View
    214

  • Download
    1

Transcript

  • Fortschr. Phys. 59, No. 7 8, 762 768 (2011) / DOI 10.1002/prop.201100025

    Cosmic strings in string perturbation theoryDimitri Skliros1,2,1 Department of Physics and Astronomy, University of Sussex, Brighton, East Sussex BN1 9QH, UK2 Cripps Center for Astronomy and Particle Theory, School of Physics and Astronomy,

    University of Nottingham, University Park, Nottingham NG7 2RD, UK

    Received 11 February 2011, accepted 13 February 2011Published online 7 March 2011

    Key words Cosmic strings, covariant vertex operators, covariant coherent states, string wavepackets.

    Fundamental cosmic superstrings are defined in string perturbation theory in terms of vertex operators,which are loosely identified with arbitrarily excited macroscopic string states. We construct explicit normalordered cosmic string vertex operators, which are given in terms of covariant coherent string states. We alsowrite down the general expression for arbitrarily excited mass-eigenstate covariant vertex operators, anddiscuss the construction of string wavepackets.

    c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

    1 OverviewCosmic superstrings are macroscopic and massive oscillating lines of energy that may have been producedin string theory realizations of the early universe, e.g. at the end of D3-D3-brane inflation, in the con-text of warped throats (KLMT) [1] or large extra dimensions [2]. The role of cosmic superstrings can beplayed by the fundamental or F-strings themselves (that define the string theory perturbatively), by the dualD-strings (or higher dimensional Dp-branes with p 1 directions compactified and 1 non-compact direc-tion), and there are also more exotic possibilities [3]. Given an initial distribution of cosmic superstrings,intercommutations/reconnections [46], string decay [79] and the possible presence of junctions [10], aswell as cosmological factors (the cosmological expansion, the distribution of matter/energy) determine thesubsequent cosmological evolution of cosmic superstrings. The upcoming gravitational wave experiments(e.g. Advanced LIGO and LISA) will be, it seems, sufficiently sensitive to detect a wide range of sig-nals from such objects [11]. The possible presence of cosmic superstrings thus provides an observationalsignature for string theory and it has therefore become of utmost importance to make sound and robustpredictions about what we expect to observe.

    The vast majority of work has (i) been based entirely on classical string evolution, thus obscuring the sig-natures unique to string theory (F-strings are only defined quantum-mechanically); (ii) backreaction effectsare systematically neglected, even though it has been shown [12] that such effects can be important evenfor order of magnitude estimates. Backreaction effects are also widely believed to be responsible for thesmall scale structure of strings, which may determine the long sought after loop production scale [1214].Furthermore, (iii) classical or supergravity computations cannot easily account for the possibility of mas-sive loops being produced, although the latter [15] may capture some strong coupling string dynamics [16].This might be expected to become significant at regions on the string where the classical evolution becomessingular (i.e. where the determinant of the embedding metric vanishes), e.g. close to cusps. The obstacles(i)-(iii) may be resolved in the context of string theory, where cosmic superstring asymptotic states are(at least perturbatively) accurately described in terms of vertex operators, V (z, z), and the correspondingcosmic superstring evolution can be rephrased in terms of S-matrix elements.

    E-mail: d.p.skliros@sussex.ac.uk

    c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

  • Fortschr. Phys. 59, No. 7 8 (2011) 763

    The S-matrix approach to cosmic string computations, pioneered in [5,7,1719], naturally incorporatesbackreaction effects (as first noticed in [7]). The above references however dealt with relatively simple andunrealistic cosmic string states, and in what follows I discuss some recent developments in the constructionof arbitrarily excited cosmic strings such a construction is expected to resolve obstacles (i)-(iii) mentionedabove, and may possibly lead to the first detections of cosmic superstrings. For further details the reader isreferred to the original articles [20] and [21].

    2 General asymptotic states

    Asymptotically free states in string scattering experiments are described in terms of vertex operators. Theseare local functionals on the worldsheet, that are composed of the fields present in the string theory; in thebosonic theory these are the spacetime embedding, X(z, z), (with = 0, . . . 25) and the worldsheetmetric, ds2 = 2gzzdzdz.1

    Let us consider the particular case of mass eigenstates and concentrate on the closed string. It is wellknown [23,24], but see also [25] and [20], that a complete set of covariant mass eigenstates can be generatedby acting with DDF operators, Ain, Ain, on a physical vacuum, e.g. eipX(z,z) with p2 = 2, for positiveintegers n. Explicitly,

    Ain =

    dz X i(z) einqX(z) and Ain =

    dz X i(z) einqX(z), (1)

    with dz dz/(2); these satisfy [Ain, Ajm] = nijn+m,0, with q a null vector, q2 0, such thatqiA

    in = 0, [Lm, A

    in] = 0 (with Lm the Virasoro generators [22]) for all n,m (similarly for Ain, Lm).

    If eipX(z,z) satisfies the Virasoro constraints [22], that is L0 eipX = eipX and Ln>0 eipX = 0 (andsimilarly for the antiholomorphic sector), general mass eigenstate vertex operators are given by:

    V (z, z) = CU(z)U(z), (2)

    which is guaranteed to be physical and covariant when [20],

    U(z) = i...jAin1 . . . Ajnge

    ipX(z)

    = : i1...igg/2a=0

    Sg/

    a=1

    i(21)i(2) Sn(21),n(2)

    gq=2a+1

    Hi(q)n(q) e

    ikX(z) :(3)

    with g/2 indicating that the maximum value of a saturates the inequality a g/2. The quantities Hin(z)and Sn,m(z) are defined in terms of elementary Schur polynomials, Sn(z),

    Hin(z) = piSn(z) +

    =1i

    (1)! X i(z)Sn(z),

    Sm,n(z) =n

    r=1 rSm+r(z)Snr(z),

    1 An arbitrary worldsheet can be parametrized by the metric ds2 = 2gzzdzdz, and because the theory is conformal [22], wecan always locally further set, say, gzz = 1/2. We use Euclidean coordinates where z = ei(+i), z = ei(i) [22] with [0, 2), and . Furthermore, we use units where = 2

    www.fp-journal.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

  • 764 D. Skliros: Cosmic strings in string perturbation theory

    which in turn, writing as 1s! inq sX(z) (and0dw/w = i), read:

    Sn(z) = i0

    dw wn1 expn

    s=1

    asws, (4a)

    =

    k1+2k2++nkn=n

    ak11k1!

    . . .aknnkn!

    (4b)

    with the properties Sn

  • Fortschr. Phys. 59, No. 7 8 (2011) 765

    This removes the N -dependence from the -function, (k2 + 2N 2) = (p2 2), and dk dk+ =dp dp+. Ignoring the tachyon, so that (k0) = (p+), the Lorentz invariant phase space reads,

    d26k

    (2)26(2)(k2 + 2N 2)(k0) =

    d26p

    (2)26(2)(p2 2)(p+)

    =

    R24

    d24p(2)24

    0

    dp+

    21

    2p+,

    (8)

    where we have integrated out p, so that p = 12p+ (p22), the tachyon onshell condition. Next introduce

    a quantity, (p) (defined in (10)), and construct a string wavepacket as follows (U is not to be confusedwith U in (3)),

    U(z, z) 1T

    d24p(2)24

    dp+

    2(p)2p+

    V (z, z). (9)

    One is to take the formal limit T in scattering amplitude computations. The change of integrationvariables (7) was such that p coincides with the momentum of the tachyonic vacuum on which DDFoperators act, and q = 1/p+. As the Lorentz invariant measure is independent of N , we may identifyV (z, z) with a mass eigenstate (2) or (see below) a coherent state vertex operator (11), both of which aredefined on the mass shell. On account of (6), we learn that,

    U

    |U = {ni},{ni}

    d24p(2)24

    dp+

    2(p)(p)(2p+)2

    ,

    d24p(2)24

    dp+

    2|(p)|2(2p+)2

    1, (10)

    where the second expression is to be thought of as the definition of the function (p) (equivalently (p))and ensures that wavepackets have unit norm, U|U = 1.

    3 Cosmic string asymptotic states

    Let us now discuss how to apply the above construction to the case where the asymptotic states of interestare cosmic strings. Loosely speaking, we define:

    cosmic (super)string macroscopic string state.

    We could also become more specific and demand that the string state be quasi-classical, i.e. that expec-tation values should evolve classically with small uncertainties, provided these are compatible with thesymmetries of the theory. The quasi-classicality requirement may or may not be relevant for cosmic stringevolution, or it may be that string states become quasi-classical if they are macroscopic (as is often as-sumed [5]). In practice, of course, it is also necessary that this object be produced in the early universe,possibly in a phase transition after a period of inflation, as is the case for instance in the context of braneinflation scenarios of the early universe, see e.g. [28].

    Appealing to our general intuition about harmonic oscillator coherent states [29], natural candidate cos-mic string states are then string coherent state vertex operators.4 The construction of covariant coherentstates in string theory has been a mystery since the 80s [30,31], and only recently has it become clear howcoherent states can be constructed in covariant gauge [21].

    4 Terminology: a better term is possibly string coherent vertex operators, or coherent vertex operators for short.

    www.fp-journal.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

  • 766 D. Skliros: Cosmic strings in string perturbation theory

    Given that we know how to construct arbitrarily excited covariant vertex operators (Sect. 3) we proceedby analogy and write down the following candidate string coherent vertex operator,5

    V(z, z) = C 20

    ds exp{

    n>0

    1n

    einsn An +m>0

    1m

    eimsm Am}

    eipX(z,z) (11)

    with C =[ 2

    0ds exp(

    n>0

    1n |n|2eins + 1n |n|2eins)

    ]1/2a (combinatorial) normalization con-

    stant. The polarization tensors, {in} for n Z and i = 1, . . . , 24, are defined to satisfy, n q = 0,n = n, and

    n=1 |n|2 < , and similarly for the anti-holomorphic sector {in}. Given that the

    DDF operators are proportional to einqX+(z) or eimqX+(z), with n,m Z+ (see comments below (6)),V(z, z) is an eigenstate of pi and p+, but not of p. The normalization is such that:

    V |V

    = T (2)(p

    + p+)(2)2424(p p)N , (12)

    with N CC 20

    ds exp(

    n>01n

    n n eins + 1n n n eins

    ) (and N = 1). These verticesform a complete set [20],

    1 =1T

    dp+

    2d24p(2)24

    (n,i

    d2in2n

    )(n,i

    d2in2n

    )VV, (13)

    with 1 the unit operator with respect to the Hilbert space spanned by V, so that 1 V = V. Onaccount of (3) and (11), one can immediately write down the normal ordered coherent vertex operators.

    The corresponding coherent state wavepacket representation, U

    (z, z), is obtained from (9), withV (z, z) V(z, z), and (10) with {ni},{ni} N . Note that U

    has unit norm: U

    |U

    = 1.

    Coherent states have classical expectation values. In the wavepacket representation, we find p U

    |p|U

    = p

    n |n|2q, with

    . . .

    d24p(2)24

    dp+

    2|(p)|2(2p+)2

    . . . , and 1 1.

    If (p) is peaked around, say, p0 , then p = p0 , (recall that q = 1/p+). Note also, that L0 L0 isthe generator of spacelike shifts, [L0 L0, X ] = iX and that vertex operators are invariant under suchshifts, (L0 L0) V = 0. It follows that it is not possible [21, 31] for a (lightcone or covariant) state tosatisfy

    X(z, z) x = (X(z, z) x)

    cl, unless [21] we work in lightcone gauge and spacetime is

    lightlike-compactified.6 Rather, an alternative requirement for a state with a classical interpretation is (inthe wavepacket representation):7

    :[X(z, z)x][X(z, z)x] : =

    20

    ds[X(zeis, zeis)x]

    cl

    [X(zeis, zeis)x]

    cl(14)

    where we average over -translations on the classical side. It is a simple exercise to verify (14) in lightconegauge: one takes X i(z, z) and X(z, z) to be given by the standard lightcone gauge mode expansions [22].

    5 The integral over s is not required in order for the Virasoro constraints to be satisfied [20], but is needed if spacetime is notcompactified in a lightlike direction. In particular, (pL pR) V = 0 only if the s-integral is included, with pL =

    dzX

    and pR =

    dzX. Had we not integrated over s, single-valuedness of V under spacelike worldsheet translationswould require [21]: X X + 2/p+.

    6 This is because in lightcone gauge, the constraint analogous to (L0L0)V = 0, reads p+(pL pR)V = (L0 L0 )V ,and hence lightlike compactification breaks the invariance under spacelike shifts [21]. This is a gauge problem [21, 31] andsays noting about the classicality of the underlying vertex operators [21].

    7 I thank to Joe Polchinski for suggesting this.

    c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org

  • Fortschr. Phys. 59, No. 7 8 (2011) 767

    The lightcone gauge classical evolution associated to the coherent states is then [20]:(X i(z, z) xi)

    cl= ipi ln |z|2 + i

    n=0

    1n

    (in z

    n + in zn),

    (X(z, z) x)

    cl= ip ln |z|2 + i

    n=0

    1n

    rZ

    12p+

    (nr rzn + nr r zn

    ),

    and X+(z, z)cl = ip+ ln |z|2, with p = 12p+(p2 +

    n(|n|2 + |n|2) 2

    ).

    8 We are to enforce inaddition classical level matching [20], n |n|2 = n |n|2. It follows that by choosing n, n appro-priately we can construct macroscopic string states, namely cosmic strings.

    A good overall consistency check is to show that the expectation value of the spin, S , 9 of the covariantgauge, lightcone gauge and classical descriptions are all identical, as this would provide further supportfor the conjecture that the lightcone and covariant descriptions are identical. We would expect this to bethe case if the states (11) have a classical interpretation, and because S is gauge invariant operator,[Ln, S ] = 0. We indeed find [20],

    Sijcov = Sijlc =n>0

    2nIm

    (in

    jn +

    in

    jn

    )= Sijcl , (15a)

    Sicov = Silc =m>0

    Z

    1np+

    Im(m im + m im

    )= Sicl , (15b)

    with components involving the + direction equal to zero, and diagonal components vanishing due to anti-symmetry.

    4 ConclusionsWe have described the construction of arbitrarily excited mass eigenstate and coherent state vertex oper-ators in bosonic string theory and suggest that the latter may be identified with arbitrarily excited macro-scopic cosmic string loops. We discussed their covariant and lightcone gauge realization and constructedan explicit map to the corresponding classical solutions. We also discussed the construction of stringwavepackets for both mass eigenstates and coherent states. As well as being more easy to physically inter-pret, string wavepackets correspond to more realistic string states than do the corresponding momentumeigenstates, while possessing unit norm. It will be very interesting to construct the corresponding super-string vertex operators and compute scattering amplitudes associated to cosmic string states progress inthis direction will be reported elsewhere.

    Acknowledgements I primarily thank Mark Hindmarsh for earlier collaboration and valuable conversations, andDavid Bailin and Joseph Polchinski for useful suggestions. I would also like to thank the organizers of the XVIthEuropean Workshop on String Theory 2010 in Madrid, where the majority of this work was presented.

    References[1] S. Kachru et al., JCAP 0310, 013 (2003).[2] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 429, 263272 (1998).[3] E. J. Copeland, R. C. Myers, and J. Polchinski, J. High Energy Phys. 06, 013 (2004).

    8 Here we use the definitions i0 pi , i0 pi , and pipi = p2.9 Recall that this is given by the non-zero mode contribution to the angular momentum, J =

    dzX[X]

    dzX[X], with a[] 12(a a).

    www.fp-journal.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

  • 768 D. Skliros: Cosmic strings in string perturbation theory

    [4] E. P. S. Shellard, Nucl. Phys. B 283, 624656 (1987).[5] M. G. Jackson, N. T. Jones, and J. Polchinski, J. High Energy Phys. 10, 013 (2005).[6] A. Achucarro and R. de Putter, Phys. Rev. D 74, 121701 (2006).[7] R. B. Wilkinson, N. Turok, and D. Mitchell, Nucl. Phys. B 332, 131 (1990).[8] D. Chialva, R. Iengo, and J. G. Russo, J. High Energy Phys. 01, 001 (2005).[9] R. Iengo, J. High Energy Phys. 05, 054 (2006).

    [10] E. J. Copeland, T. W. B. Kibble, and D. A. Steer, Phys. Rev. Lett. 97, 021602 (2006).[11] E. J. Copeland and T. W. B. Kibble, Proc. Roy. Soc. Lond. A 466, 623657 (2010).[12] J. M. Quashnock and D. N. Spergel, Phys. Rev. D 42, 25052520 (1990).[13] D. P. Bennett and F. R. Bouchet, Phys. Rev. Lett. 60, 257 (1988).[14] M. Hindmarsh, Phys. Lett. B 251, 2833 (1990).[15] A. Dabholkar and J. A. Harvey, Phys. Rev. Lett. 63, 478 (1989).[16] A. A. Tseytlin, Phys. Lett. B 251, 530534 (1990).[17] J. Polchinski, Phys. Lett. B 209, 252 (1988).[18] J. Dai and J. Polchinski, Phys. Lett. B 220, 387 (1989).[19] D. Mitchell et al., Nucl. Phys. B 315, 1 (1989).[20] M. Hindmarsh and D. Skliros, Covariant Vertex Operators for Cosmic Strings, hep-th/0911.5354 (2009).[21] M. Hindmarsh and D. Skliros, Phys. Rev. Lett. 106, 1602 (2011).[22] J. Polchinski, String Theory. Vol. 1: An Introduction to the Bosonic String (Cambridge University Press, Cam-

    bridge, UK, 1998).[23] E. Del Giudice, P. Di Vecchia, and S. Fubini, Ann. Phys. 70, 378398 (1972).[24] M. Ademollo et al., Nuovo Cim. A 19, 181203 (1974).[25] E. DHoker and S. B. Giddings, Nucl. Phys. B 291, 90 (1987).[26] R. Sasaki and I. Yamanaka, Phys. Lett. B 165, 283 (1985).[27] S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations (Cambridge University Press, Cambridge, UK,

    1995).[28] E. J. Copeland and T. W. B. Kibble, Phys. Rev. D 80, 123523 (2009).[29] J. R. Klauder and B. S. Skagerstam, Coherent States Applications in Physics and Mathematical Physics (World

    Scientific Publishing, Singapore, 1985).[30] G. Calucci, Nucl. Phys. B 287, 658668 (1987).[31] J. J. Blanco-Pillado, A. Iglesias, and W. Siegel, Phys. Lett. B 655, 284289 (2007).

    c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org